The purpose of this handout is to provide you with a wide assortment of applications of algebra, and an extensive bibliography if you would like to continue the 'quest for good ones'. To use it, you should know what was considered an application, and what was merely a word problem. I also would like to present some brief rationale for including more of what I call applications in the algebra curriculum.

  INTRODUCTION

What are 'applications'?

Algebraic work (in our classes) can be classified among some basic groups:

Exercises involve using patterns and examples already presented to the student. The majority of work provided in our textbooks is of this type.

To be a word problem, a exercise or problem merely needs to be stated in words -- not just in algebraic symbols. A word problem might just be an exercise, and not truly a 'problem'. Many textbooks that include the word "application" in their title or prospectus provide only word problems, not applications.

"Problem Solving" involves the student combining different skills not presented in combination, or in using concepts in a new setting. Some of our textbooks include a little of this type.

Applications involve using algebra to deal with situations in the students' lives OR to deal with occupational/professional situations that make sense to the student. Dealing with 'real objects' is not enough; the outcome must be useful or interesting to at least some students. For example, an early word problem in many beginning algebra textbooks is often like this:

Maria has a 200 m length of fence. She needs to fence a rectangular play area that has a length 5 m more than twice the width. Find the dimensions of the play area.

Although this problem deals with real objects, it is not an application. Critical students will quickly respond that the length and width are clearly already known, and we are simply trying to get them to find our answer. To be an application, the problem would be given as:

Maria can afford to buy 200 m of a certain fence. The area that she can fence can be from 30 to 40 m wide and from 40 to 70 m long.   Find the dimensions that will give her the largest fenced-in play area.

An 'application' should help answer the question "Why should I bother learning this stuff?", when answers such as "to pass the course" and "to get ready for the next course" are ruled out. Applications show how specific algebraic skills can be used to help a person live or work better.

Applications are not necessarily problem solving as defined above. Problem solving is also a valid goal of the curriculum, but applications reflect a different dimension. An application can be classified as either a word problem or as problem solving, depending on its presentation. Our textbooks historically provide very little in the way of applications.

Why should applications be included in our curriculum?

Algebra is often perceived as a subject, like latin, that one must master to get someplace else or become the 'right kind of person'. To the extent that we want students to learn and master algebra, as opposed to simply memorizing enough rules to pass, applications are needed as part of a curriculum that deeply involves the student. A vital theory of learning ("constructivism") asserts that deep learning depends on students developing their own understanding, the quality of which depends on the depth of involvement with the material.

In addition, applications offer opportunities to reinforce concepts from other fields; in this way, an algebra course becomes more diverse. A current trend in higher education curriculum is to integrate 'themes' (such as writing or critical thinking) into a variety of general education courses. My approach to these levels of algebra is that they are general education courses.

The 200 (and more) examples of applications

The examples are organized along rough topical breakdowns:

• Business (marketing, economics, agriculture, manufacturing, etc)
• Technology (construction, transportation, electrical & energy, etc)
• Life -- General (cooking, sports, consumers & finance, etc)
• Home Life -- Transportation and Safety (cars, boats, safety, travel)
• Social Science (government, education)
• Biology (medicine, wildlife, environment, etc)
  
  

The algebraic skills needed to deal with these applications range from linear equations to logarithmic functions. Many of them can be approached from either a symbolic or a graphical perspective. For example, many problems involve a quadratic relationship which can be solved symbolically or solved using a graphing utility.

The applications are presented on the next 43 pages, followed by a bibliography. Following each problem, there are two lines with this information:

For example, the first problem:

is from the domain of business, in the area of economics, uses at least 3 variables, involves an exponential equation or function, has a high student appeal (by the author's judgement), and was found in Gustafson's College Algebra text on page 373.

Much of this information is a matter of judgement. For example, many people consider economics to be a social science area. However, it is my hope that you will find this information helpful -- even if you do not agree with the details.

One final comment: Be creative and have fun! If you find something to be enthusiastic about, your students will find the problems hard to resist.

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  BUSINESS APPLICATIONS

1. In business, equipment is often depreciated using the double-declining balance method. A piece of equipment with a life expectancy of N years, costing C dollars when new, will depreciate to a value of V dollars in n years, according to . A new delivery truck is bought for $18,000, and has a life expectancy of 8 years; when will the value be $5000? What is the salvage value (when the time reaches the life expectancy)?

Business Economics 3 Variables Exponential High

Source: Gustafson COLLEGE ALG page 373

2. Wing Aero's panel stamping machine was worth $32000 new. It is now 6.5 years old, and it is worth $12000. Compare exponential and linear models for its depreciation. Which model is better if the expected lifetime of the machine is 15 years?

linear model: V=32-3.07Y; exponential: V=32(0.86)^Y, where V is in thousands in Y is the number of years since purchase.

Business Economics Two Variables Exponential High

Source: adapted from Bolker USING ALG page 127

3. A certain factory has 3000 light bulbs. It was determined that if all the light bulbs were replaced at essentially the same time, the number of bulbs which had burned out after t hours was . (Such functions can be used to show it is more economical to replace all light bulbs at once rather than just as they burn out.) How many bulbs have burned out after 100 hours, 500 hours, and 900 hours? If you want to replace all bulbs after 35% of them have burned out, when should you do this?

Business Manufacturing Two Variables Exponential High

Source: Hestenes ALG AND TRIG pages 218-219

4. A newspaper had 13,440 subscribers in 1991, and 14,112 in 1992. If the rate of growth remains constant, what will the number of subscribers be in 1996?

Business Marketing Two Variables Exponential High

Source: Rotman

5. A general form of the learning curve is , where P is the performance, M is the maximum performance, B and r are constants for a given learner and a given learning situation, and t is the time. The formula reflects the fact that learning is more rapid in the initial periods, and the rate of learning slows as the performance approaches the maximum. In one situation, an average new employee is initially tested and was found to have a score of 31 out of a maximum of 100. Following two weeks of training, her score improved to 53. Use these values to find the equation of this learning curve relating P to the time in days. Also, find the time needed to earn a score of 80. Compare the time needed to achieve scores of 90 and 95. Do you see why learning to perfection is not always a good goal?

Business Other Two Variables Exponential High

Source: LCC

6. A farmer wants to plant a combination of two crops, cabbage and corn, on 100 acres. Cabbage requires 60 man-hours of labor per acre, and corn requires 80 man-hours of labor per acre. If the farmer has 6600 man-hours available, how many acres of each crop should be planted?

Business Agriculture Two Variables First Degree High

Source: Piascik COLLEGE MATH page 248

7. A company has fixed costs of $100 per week, and each item it produces costs $10 to manufacture. The company sells each item for $12. Find the number of items the company must sell in order to break even.

Business Economics Two Variables First Degree High

Source: McKeague Interm pg 485

8. A small manufacturing company produces tables and chairs in one part of its operation. Each chair requires 3 feet of lumber, and each table requires 7 feet of lumber. A chair requires 2 hours of labor, and a table requires 8 hours of labor. The profit on a chair is $1, and the profit of a table is $3. In a given week, there are 420 feet of lumber and 400 hours of labor available. How many of each item should the company produce in order to make its profit as large as possible?

Business Manufacturing Two Variables First Degree High

Source: Sloyer FANTASTIKS page 24

9. Two types of carburetors are being assembled. Each type A carburetor requires 15 min of assembly time, whereas type B carburetors require 12 min each. Type A carburetors require 2 min of testing whereas type B carburetors require 3 min each. There are 222 min of assembly time and 45 min of testing time available. How many of each type should be assembled and tested if all of the available time is to be utilized?

Business Manufacturing Two Variables First Degree High

Source: Washington, page 432

10. Let us suppose that there are four machines in a certain factory located along an aisle (shown as a number line) as illustrated:

         {a number line graph, with points A, B, C, and D marked at coordinates -3, -1, 2, 6 }

A new machine is to be located somewhere along this aisle. There will be equal interaction between all of the existing machines and the new machine, since it will supply materials. Assuming that the cost for moving parts only depends upon the distance, where should the new machine be located to minimize the cost of moving the materials?

Variation: Machine A uses twice as many items as machine B, and machine C uses three times as many items as machine B. Where should the new machine be located?

Business Manufacturing One Variable First Degree High

Source: Sloyer FANTASTIKS page 61

11. A fast food restaurant should be put in a densely populated area on a heavily traveled street. Experience shows that the weekly gross sales G (in thousands of dollars) can be predicted by the formula G = 0.373P + 0.157C, where P is the population in thousands living within 3 miles of the restaurant and C is the number of cars, in thousands, which pass the restaurant each day. Predict the gross sales for a restaurant on a road with 7200 cars each day and with 10100 people within 3 miles.

Business Marketing Two Variables First Degree High

Source: based on Bolker USING ALG page 78

12. A salesperson earns 16% commission on sales over $2,000. What sales are needed for earning $3,000?

Business Retail One Variable First Degree High

Source: MTH 102 applications supplement 1

13. A company manufactures open trash bins. The bins must be 6 feet high and have a volume of 192 cubic feet. The material for the front and back costs $5 per square foot; the material for the sides costs $10 per square foot; and the material for the bottom costs $20 per square foot. (These differences are due to the thicknesses required to be strong enough.) Write an expression for the total cost as a function of the length and width. Use the fact that V = LWH to write the cost in terms of just one other variable; graph this function. Using the graph, find the width and length of the bin that would minimize the total cost per bin.

Business Manufacturing 3 Variables Rational High

Source: adapted from Piascik COLLEGE MATH page 558

14. A computer, using data from a refrigeration plant, estimated that in the event of a power failure, the temperature T, in degrees Celsius, in the freezers would be given by: where h is the number of hours after the power failure. How long would it take for the temperature to reach 0C?

Variation: Some products require the temperature to be below -10C for safe storage; when the temperature goes above this level, the products have to be sold as damaged. How long would it take to reach this temperature?

Business Other Two Variables Rational High

Source: Washington BASIC TECH MATH page 77

15. If an apple grower harvests his crop now, he will pick, on the average, 120 pounds per tree. He will get $0.48 per pound for his apples. From past experience, he knows that for each additional week he waits, the yield per tree will increase by about 10 pounds, while the price will decrease by about $0.03 per pound. Write an equation for the total revenue related to x, the number of weeks he waits. How many weeks should the grower wait in order to maximize sales revenue?

Business Agriculture Two Variables Second Degree High

Source: Piascik COLLEGE MATH page 91

16. From experiments, the yield per apple tree in bushels is estimated by: y = -3t + 240 (where t30); y is the yield per tree, and t is the number of trees per acre. As a consequence, the total yield is given by: Y = -3t² + 240t. Find the total yield if there are 30 trees per acre. Find the maximum total yield, and the number of trees per acre for the maximum yield.

Business Agriculture 3 Variables Second Degree High

Source: adapted from Farlow APPLIED MATH page 607-608

17. An oil well is located on a side road 1.2 km from the main road. The storage tanks are on the main road, 2.5 km from its intersection with the side road. A pipeline is to be built connecting the well to the tanks. It costs $3000 per km if it is built along the roads or $4000 per km if it is built directly across the fields to the tanks. Which way is cheaper, and by how much?

Business Economics One Variable Second Degree High

Source: adapted from Foerster Algebra I pg 551

18. Base-M Co introduces new product, and wishes to make $200,000 in yearly profits. The yearly demand function is d = -20s + 6000, where s is the selling price; the profit per unit is given by p = s - 50. What selling price should they establish?

Business Economics 3 Variables Second Degree High

Source: MTH 102 applic supp 4

19. MacKenzie Park sells its trivets for $0.25 per unit and during 1969 has net sales of $500,000 and a net income of $35,000. Production capacity is limited to 15,000 trivets per day and trivets are produced 300 days each year. The variable cost (to manufacture each trivet) is $0.10 per trivet. The trivets are made in lots of 300. The wholesaler finds that the number of defective trivets in each lot of 300 is equal to the daily unit production rate divided by 200. (That is, the more produced, the higher the rate of defectives.) The company reimburses the wholesaler $0.50 for each defective trivet it finds. Write an expression for the number of defective units per day. Write an equation that gives the profit per day. Use this equation to find the number of units that will give the maximum profit per day.

Business Marketing Two Variables Second Degree High

Source: adapted from Piascik COLLEGE MATH pages 87-88; he cites the problem as having been on a previous Uniform CPA Examination)

20. As the operational director for a chain of family restaurants, you need to decide on the best price for the popular Family Platter, which is now $8.00. Currently, 250,000 Family Platter meals are sold per week. Company records and market research yield the following formulas:

Find the price that will give you the maximum profit per week. If this is not a typical price (a multiple of $.25), find the best price. Also, what will the volume of meals be at this best price? What will the profit be?

Business Marketing 3 Variables Second Degree High

Source: adapted from Farlow APPLIED MATH page 604

21. Rhea has a home day care business, and needs to have a fenced play area for the children she cares for. She has 150 feet of fencing, and needs to have at least 1500 square feet if play area; she doesn't want any more than that so the rest of the yard can be used for other activities. What dimensions should the rectangular play area be?

Business Other Two Variables Second Degree High

Source: based on UCSMP Adv Algebra pg 723

22. The Sharp Company manufactures table knives. Each knife costs $12 to produce and sells for $16. The quality control manager has determined from past data that the equation d = x²/(20,000,000) relates the fraction of defectives (d) produced with daily production volume (x). (Note that the values of d range from 0 to 1.) Each defective knife costs the company an additional $20. Determine an equation that expresses daily profit as a function of daily volume. What daily volume produces the maximum daily profit?

Business Marketing Two Variables 3rd Degree High

Source: adapted from Piascik COLLEGE MATH page 99

23. Let us suppose that the yield y of tomatoes in bushels per acre increases depending on the number x pounds of fertilizer per acre. For one situation, the relationship is .

Graph this function, and identify the maximum yield (approximately) per acre. Challenge: If tomatoes sell for $6.00 per bushel and fertilizer costs $0.80 per pound, how many pounds of fertilizer per acre should be applied to yield the maximum profit per acre? If we take labor of harvest into account ($0.50 per bushel), what fertilizer application will yield the maximum profit per acre?

Business Agriculture Two Variables Exponential Medium

Source: adapted from Varberg APPLIED CALC page 201

24. Graph the relation for P = $1000, r = 7% and n = 12. Using this graph, estimate the number of years needed to have $2700 in the account.

Business Economics 3 Variables Exponential Medium

Source: Rotman

25. The costs of a manufacturing process are divided between labor costs L and capital costs K (raw materials, equipment, and so on). In many cases, economists find that a Cobb-Douglas production function provides a good model for the number of units F that will be produced. For a certain manufacturing process, the Cobb-Douglas function is believed to be

How many units of output will result from inputs of 64 units of labor and 27 units of capitol? Which will increase output more, doubling the number of units of labor or doubling the number of units of capital?

Business Manufacturing 3 Variables Exponential Medium

Source: Varberg APPLIED CALC page 313

26. The number of subscribers to cable television t months after its introduction in a certain city is expected to be

How many subscribers will there be after 6 months? How many subscribers will eventually exist? (Hint: try a few large values for the time -- 5 years, 7 years, 10 years, etc.)

Business Marketing Two Variables Exponential Medium

Source: from Berkey APPLIED CALC page 245

27. Oil engineers have started pumping gas from a new well in the Gulf of Mexico. On the basis of preliminary tests and past experience they predict that the monthly production of gas p after t months since pumping begins will be given by the function:

, where p is measured in millions of cubic feet.

Find the expected oil production for each of the first 6 months. Find the expected oil production for months 24 to 30. After 20 years, what will the monthly oil production be?

Business Marketing Two Variables Exponential Medium

Source: based on Farlow APPLIED MATH pages 771-772

28. The demand, x, (in millions), for a certain product is related to its unit price, p (in dollars), by the equation:

.

Solve this equation for demand in terms of unit price. What will be the demand when the unit price is $6?

Variation: The linear model for demand would be x = 20 - 0.55p. Compare the linear and logarithmic models; in particular, compare the demand they each predict for small prices ($1 or less), and large prices ($20 or more).

Business Marketing Two Variables Exponential Medium

Source: adapted from Piascik COLLEGE MATH pages 151-152

29. Businesses often measure training programs by how fast new employees can learn new skills. Using a given training program, suppose the average number of units L produced by a new employee during week t can be described by the exponential learning curve . Find the number of units an average employee can produce during weeks 2, 4, 5, 6, and 8. Draw a graph of this learning curve.

Business Other Two Variables Exponential Medium

Source: adapted from Farlow APPLIED MATH page 662

30. A rule-of-thumb used by car dealers is that the trade-in value of a car decreases by 30% each year, so that the value of the car at the end of a year is 70% of what it was at the start of the year. Find the exponential model for the trade-in value of a car that had a trade-in value of $5400 originally. What will the trade-in value be after 4 years? In how many years would the trade-in value of the $5400 car decrease to $1200?

Business Retail Two Variables Exponential Medium

Source: adapted from Foerster ALGEBRA & TRIG page 166

31. A distributor of hybrid corn has two storehouses, A and B. At A is stored 110 tons of corn; 190 tons are stored at B. Farmer X ordered 60 tons to be delivered to his farm, and farmer Y ordered 80 tons for her farm. The shipping costs appear in the following table.

How should the orders be filled to minimize the shipping costs?

Business Agriculture Two Variables First Degree Medium

Source: Gustafson COLLEGE ALG page 273)

32. There are 925 seats in an auditorium. For a concert, student tickets will sell for $5 each, while regular admission is $8. If the income from the tickets must be at least $6000, how many of each kind of ticket could be sold?

Business Marketing Two Variables First Degree Medium

Source: Rotman

33. A small business started the day with $250, ended the day with $1204, and tax rate is 6%. How much should she send in for sales tax?

Business Retail One Variable First Degree Medium

Source: McKeague Interm pg 135

34. A retailer wants to sell an item for exactly $10 including sales tax. If the sales tax rate is 5%, how much should he or she charge for the item without tax?

Business Retail One Variable First Degree Medium

Source: UCSMP Algebra, pg 288

35. In business, equipment is often depreciated using the double-declining balance method. A piece of equipment with a life expectancy of N years, costing C dollars when new, will depreciate to a value of V dollars in n years, where n is given by . A computer with an expected life of 5 years costs $37000. It has depreciated to $8000. How old is it?

Business Economics 3 Variables Logarithmic Medium

Source: Gustafson COLLEGE ALG page 373

36. Based on a study of potato yields, a biologist has obtained the formula where y is the yield per plant (kg) and r is the density (plants per hectare); r100. Write an equation giving the yield per hectare. Graph both functions; explain why the yield per plant keeps decreasing while the yield per hectare increases.

Business Agriculture 3 Variables Rational Medium

Source: from Varberg APPLIED CALC page 74

37. The precipitation rate of a sprinkler system is given by the formula: ,   where P = precipitation rate (inches/hour) , F = flow rate (gal/min), S = spacing between sprinklers (ft) , L = spacing between rows (ft)

A farmer needs a precipitation rate of 10 inches/hour, and knows that the flow rate is 30 gal/min. If S = L, find the sprinkler spacing.

Business Agriculture 3 Variables Rational Medium

Source: adapted from Saunders WHEN ARE WE EVER GONNA page 108

38. The ABC Container Company manufactures plastic bottles. Its total cost function is defined by C = (x - 5)³ + 1025, where x is the number of bottles produced (in millions) and C is the total cost (in thousands of dollars). The average cost (per bottle) is: . Graph the total cost and average cost on the same coordinate system; you can do this by choosing several values for x and then computing the total cost and the average cost. (Or, use a graphing utility or calculator.) From the graph, estimate the number of bottles that would minimize the average cost. (Note that this does not find the production to maximize total profits.)

Business Marketing 3 Variables Rational Medium

Source: adapted from Piascik COLLEGE MATH page 546

39. A telephone company estimates that the number N of phone calls per day between two cities of population P₁ and P₂ that are d miles apart is given by the equation:

Estimate the population of the second city if the first city has a population of 48000, the distance is 75 miles, and the phone circuits can handle 1,100,000 calls per day.

Business Marketing 3 Variables Rational Medium

Source: Aufmann INTRO ALG page 90

40. A company produces tires for trucks. The total production cost for producing x tires is given by C = 50x + 30,000. The average cost is given by dividing the total cost by the number of tires produced. Write a function for the average cost. What is the average cost for producing 100 tires and for 1000 tires? Graph this function. As the number of tires produced gets very large, the average cost approaches what value?

Business Marketing 3 Variables Rational Medium

Source: adapted from Piascik COLLEGE MATH pages 116-117

41. In forestry, a formula used to determine the volume V of a log is V = ½L(B + b), where L is the length of the log, and B and b are the areas of the ends. Find b if V = 38.6 cubic feet, L = 16.1 ft, and B = 2.63 ft².

Business Agriculture 3 Variables Second Degree Medium

Source: Washington BASIC TECH MATH page 42

42. An insurance company predicts the number of accidents per 50 million miles driven by the formula A = .4x² - 36x + 1000, where x is the age of this group of drivers. What age drivers would be predicted to have 200 accidents per 50 million miles driven?

Business Economics Two Variables Second Degree Medium

Source: MTH 102 applic supp 4

43. When a popular brand of CD player is priced at $300 per unit, a stereo store sells 15 units per week. Each time the price is reduced by $5, however, the sales increase by 1 per week. What selling price will result in weekly revenues of $7000?

Business Marketing Two Variables Second Degree Medium

Source: Swokowski, pg 84

44. A publisher decides that, for aesthetic reasons, the pages of a book should have margins of 1½ inches on top and bottom, and 1 inch on each side. A psychologist advises the publisher that the optimal amount of printed matter per page is 35 in². Paper is rather expensive, so the publisher seeks to minimize the amount of paper used. Write a function for the total area of the page in terms of the length of a page. Using a graphing utility, find the minimal area. If the ratio of width to length should be 0.75, find the dimensions the publisher should use.

Business Marketing Two variables Second Degree Medium

Source: based on Faber APPLIED CALC page 128


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  TECHNOLOGY APPLICATIONS

45. A certain satellite has a power supply whose output in watts is given by the equation where t is the number of days the battery has operated. If it is operated continuously after the satellite was placed into orbit, how many watts is the battery putting out after 1 year? If it takes at least 10 watts to operate the satellite, how many days can the satellite be used?

Technology Electrical Two Variables Exponential High

Source: Hestenes

46. The current charge C of a battery being charged at time t follows this formula:

(M is the theoretical maximum charge and k is a positive constant that depends on the battery and the charger. This formula shows that the charge increases quickly at first, but the rate of charging slows dramatically as the charge approaches the maximum. If a battery can reach half of its full charge in 6 hours, how long will it take the battery to reach a 90% charge? a 99% charge? (We assume that the battery was fully discharge when it began charging.)

Technology Electrical 3 Variables Exponential High

Source: adapted from Gustafson COLLEGE ALG pages 371 & 373

47. A truck averages 35 mph for the first four hours of a trip. How long must the truck travel at 55 mph to have an overall average speed of 50 mph?

Technology Transportation Two Variables First Degree High

Source: UCSMP Algebra pg 563

48. On November 24, 1973 a Los Angeles Times Service bulletin stated that Alan Glassenapp, staff engineer in the GMC Division of Research and Development, insisted that "big trucks and buses will get an average of 5 percent fewer miles per gallon traveling at 55 mph than at 50." He also stated that "in terms of cost effectiveness, the higher speed makes sense." Assuming that fuel costs $1.10 per gallon and that a typical truck gets about 8 miles per gallon at 50 mph, show how the second statement can be true when the first one is also true. (Note that there are three basic costs for the truck: fuel, driver's hourly wage, and other operating costs.)

Technology Transportation 3 Variables First Degree High

Source: Sloyer FANTASTIKS page 13

49. A certain Sikorsky helicopter can carry cargo internally within its fuselage or externally slung beneath its fuselage. Carrying cargo externally reduces the flight speed, but also reduces the loading and unloading time. The specifics are:

For what distances is the external load preferred?

Technology Transportation One Variable First Degree High

Source: Sloyer FANTASTIKS page 9

50. A cargo plane is ready to be loaded. There is first class freight to be loaded weighing 11,300 lb. There are also boxes labeled 'ship when possible' weighing 80 lb each. How many boxes can be added within load limit of 15,000 lb?

Technology Transportation One Variable First Degree High

Source: MTH 102 applic supp 1

51. A bridge across a gulch ('very small valley') is held up by a series of circular columns 20 inches in diameter; the columns have a varying length, with the longest ones in the middle of the span. From a Strength of Materials text you find that as you put more and more weight on a column, it can collapse either by buckling (breaking outward) or by crushing (compression). The load which will buckle a column is directly proportional to the fourth power of its diameter and inversely proportional to the square of its length. The load which will crush a column varies directly with the square of its diameter and is independent of its length. A laboratory test shows that a 2 inch diameter column of the same material used in the bridge 3 feet long will buckle under a load of 4 tons. The same column will be crushed by a load of 5 tons. Write equations predicting the buckling load and the crushing load for this type of column. Predict the buckling and crushing loads for the 20-inch columns under the bridge that are 20 feet, 30 feet, and 50 feet long. Estimate the length of columns on the bridge that would collapse by BOTH buckling and crushing.

Technology Construction 3 Variables Rational High

Source: from Foerster ALGEBRA & TRIG pages 240-241

52. The crushing load for a square pillar is given by the formula L = crushing load (tons); T = thickness of wood (inches); H = height of post (ft).

Find the crushing load for a 4-inch post 8 feet high.

Variation: A post needs to be 12 feet high and support 80 tons. What thickness should be used?

Technology Construction 3 Variables Rational High

Source: Saunders WHEN ARE WE EVER GONNA page 98

53. A streetlight is situated on a pole x feet above the ground. The intensity of illumination I at a point P that is 24 feet from the base of the pole is known to vary according to the equation , where d is the distance from P to the light and k is some positive real number. Write a function relating the illumination to k and x; graph this function for any convenient value of k. How high above the ground should a security light be placed to provide maximum illumination at the point P on the ground?

Technology Electrical Two Variables Rational High

Source: adapted from Demana COLLEGE ALG page 261

54. A highway engineer has sampled the speed and density of automobiles along a particular section of highway and determined that a good model of the relationship between velocity, density, and traffic flow is the set of equations:

    q = pv

where v is the velocity (km/h); p is density (in 100's per km); and q is the flow (in 100's per hour). If the traffic flow needs to be 50 100's per hour, what should the speed limit be?

Technology Transportation 3 Variables Rational High

Source: adapted from Berkey APPLIED CALC pages 170-171

55. The cost of operating a certain truck (fuel, oil, and so on) is 40 + 0.25s cents per mile when driven at s miles per hour. The driver gets $12 per hour. Two trips need to be made, one of 200 miles and the other of 1000 miles. At what speed should the truck be driven on each trip to minimize the total cost?

Technology Transportation 3 Variables Rational High

Source: adapted from Varberg APPLIED CALC page 139

56. Two surveyors with two-way radios leave the same point at 9:00 am, one walking due south at 4 mph and the other due east at 3 mph. How long can they communicate with one another if the radios have a maximum range of 2 miles?

Technology Construction Two Variables Second Degree High

Source: Swokowski, pg 83

57. At the scene of an accident, the skidding distance was 200 ft; police used formula V² = 30FS, where V is mi/hr, S is feet, and F is a decimal value of a friction coefficient. For this road, the friction coefficient is 0.60. The speed limit was 45 mi/hr. Was the car speeding?

Technology Transportation 3 Variables Second Degree High

Source: MTH 102 applic supp 4

58. Police officers know that a car's speed and the road conditions affect a driver's ability to stop quickly. The braking distance is the distance traveled by a car from the instant the driver applies the brakes until the car comes to a complete stop. The following formula can be used to estimate braking distance:

where b is the estimated braking distance in feet, r is the car's speed in miles per hour, and F is the driving surface factor, given by the table below.

What braking distance would you estimate for a car traveling on dry asphalt at a speed of 55 miles per hour?

At the scene of an accident, a car's skid marks indicate that it required about 215 feet to come to a complete stop on wet asphalt. Find an estimated speed for the car before it began braking.

Extension: You are involved in deciding how a new road should be paved. One factor in deciding is the safety issue relating to the stopping distances of cars. The local weather service office indicates that you can expect the road to be wet about 25% of the time. Should the road be asphalt or should it be concrete?

Technology Transportation 3 Variables Rational High

Source: adapted from CORD APPLIED MATH

59. In planning a domed baseball stadium, engineers need to know how high a ball might be hit. This equation describes the maximum height of a ball when it is x feet from home base: h = -.005x² + 2x + 3.5 (h and x are in feet). How high will the ball get, and how far from home plate will this be? The height of the ball is related to time since it was hit by the equation h = 113t - 16t² + 3.5. How long would the ball be in the air from when it was hit until it reaches the ground? How far from home plate would the ball land on the ground?

Technology Construction 3 Variables Second Degree High

Source: based UCSMP Adv Algebra pg 344

60. The force F on a vertical surface varies jointly as the area A of the surface and the square of the wind speed S, and also depends on the kind of material being used. For a particular kind of window, the formula is: The breaking force for the window is 500 lb, and it has a surface area of 20 square feet. What is the maximum wind speed that it can withstand before it breaks?

Technology Construction 3 Variables 3rd Degree High

Source: adapted from UCSMP Adv Algebra page 113

61. The consumption of fuel, c, in gallons per hour of a certain engine is determined as a function of the number r, in revolutions per minute r/min, of the engine to be: c = 0.0029r + 1.06. This formula is valid from 500 r/min to 3000 r/min. Plot c as a function of r. Estimate the r/min if the fuel consumption is 7.5 gal/hr.

Technology Other Two Variables First Degree Low

Source: Washington BASIC TECH MATH page 66

62. Furnaces are bought by the amount of heat they can add to a house. A formula can be used: Q = UA(T - M), where

Q = BTU per hour of heat needed from the furnace     U = coefficient of heat transfer for the outside surfaces, which varies with the material used

A = surface area (square feet)      T = desired room temperature (F)      M = minimum outside temperature (F)

Find Q if the surface area is 3250 square feet, U = 0.0679, the desired temperature is 68 F, and the lowest outside temperature is -6 F.

Variation: How many square feet of surface could a house have if Q = 30000 BTU, U = 0.0725, the desired temperature is 70 F, and the lowest outside temperature is 10 F?

Technology Construction 3 Variables 3rd Degree High

Source: adapted from Saunders WHEN ARE WE EVER GONNA page 112

63. The force on the blade of a wind generator varies directly as the product of the blade's area and the square of the wind velocity. For a particular blade design, the force was 19.16 lb when the area was 3.72 ft² and the wind was 31.38 ft/sec in a wind tunnel. To support the electricity requirements of a certain house, 30 lb of force is needed. It is known that winds of 15 mi/hr are typical for the region. What blade area is needed?

Technology Electrical 3 Variables 3rd Degree High

Source: adapted from Washington, page 170

64. A large suburban community receives its electrical power supply from an urban power plant which burns coal. A local engineer, after studying data on energy usage, determines that a good model for daily energy usage between 6:00 am and 8:00 pm is given by:

where 6 <= t < = 20.

The engineer argues that, through the use of solar generators, the community could reduce its energy demand from the urban power plant by the amount S, where S is given by:

.

Use a graphing utility to graph each function. Write a function for the resulting energy usage R if solar generators are used; graph this function on the same coordinator system. Use these graphs to identify the maximum reduction, and the difference between the old peak demand and the new peak demand.

Technology Electrical 3 Variables 3rd Degree High

Source: adapted from Berkey APPLIED CALC page 166

65. The intensity of light through ordinary glass of thickness t (in centimeters) is modeled by the exponential equation where is the intensity before entering the glass. How thick must the glass be to block out 10% of the light?

Variation: Typical house windows have two panes of glass that are each about 0.4 cm thick. What percent of the outside light reaches the inside of the house?

Technology Construction 3 Variables Exponential Medium

Source: UCSMP Adv Algebra page 531

66. Estimates of vertical wind shear are of great importance to pilots during take-offs and landings. Since it is impossible to know the wind speed at every height, the wind shear must be estimated by using only a finite number of ordered pairs. An estimation of the wind shear can be found with the formula:

, where the v's are the wind speeds at altitudes represented by the h's. There is also an empirical relation between the wind speeds and the altitudes: , where p is a fractional exponent.

Suppose that at a height of 20 feet above the ground the wind speed is 28 mi/hr. With a typical value of p = , estimate the vertical wind shear 200 feet above the ground.

Technology Transportation 3 Variables Exponential Medium

Source: from Swokowski CALCULUS page 134

67. The electric resistance R, in ohms, of a resistor change with temperature according to the formula: is a constant for a given type of resistor and R₀ is the resistance at 0C. For given resistor, R₀ = 712 ohms and = 0.00455/C. Determine the value of T so R will have a value of 825 ohms.

Technology Electrical Two Variables First Degree Medium

Source: Washington BASIC TECH MATH page 42

68. Millie Watt is an electrical contractor. She needs to find out how many feet of wire are left on a partially used roll, without having to unroll the wire and measure it. She cuts off a 4-foot piece, and finds that it weighs 1.2 pounds. The roll of wire remaining weighs 36 pounds. How many feet of wire are there?

Technology Electrical Two Variables First Degree Medium

Source: adapted from Foerster Algebra I pg 576

69. A jet airplane uses a fixed quantity of kerosene for a takeoff and a landing (combined) and, when in the air, a certain quantity per mile. If a 400-mile trip requires 1455 gal and a 260-mile trip uses 1077 gal, find the fuel used for a takeoff and landing, and the rate of fuel consumption during flight. How much kerosene will a 1230-mile trip require? If the jet has tanks that can hold 4290 gal, how long of a trip can it make? (note: model is F = 375 + 2.7M)

Technology Transportation Two Variables First Degree Medium

Source: Bolker USING ALG page 25 & 31)

70. Greenshield's formula can be used to determine the amount of time that a traffic light at an intersection should remain green. This formula is: G = 2.1n + 3.7, where G is the green light time (seconds) and n is the average number of vehicles traveling in each lane per light cycle. Find the green time for a traffic signal on a street that averages 19 vehicles in each lane per cycle.

Also, you notice that a particular light has a green time of 25 seconds; about how many vehicles per lane go through the intersection?

Technology Transportation Two Variables First Degree Medium

Source: based on CORD APPLIED MATH

71. The maximum load L a horizontal beam can safely hold varies jointly with the width w and the square of the depth d and inversely with the length l. If a 10-foot beam with width 3 and depth 4 will safely hold up to 800 pounds, how many pounds will a 12-foot beam with width 3 and depth 4 hold?

Variations: What width is needed to support 1000 pounds?

Technology Construction 3 Variables Rational Medium

Source: McKeague Interm pg 452

72. An individual is interested in building a rectangular one-story house containing 180 square meters of floor space. Due to energy conservation considerations, the perimeter of the house is to be as small as possible. Also, for aesthetic reasons, neither the length nor the width of the house is to be less than 10 meters. Express the perimeter of the house in terms of its length. Graph this relationship, and find the best dimensions for the house.

Technology Construction Two Variables Rational Medium

Source: from Berkey APPLIED CALC page 167

73. An engineer plans to build a tunnel through a hill. The arch of the tunnel will be in the shape of a parabola, and the tunnel will span a two-lane highway that is 8 meters wide. To allow safe passage for most vehicles, the tunnel must be 5 meters high at a distance of 1 meter from the tunnel's edge. What will be the maximum height of the tunnel?

Technology Construction 3 Variables Second Degree Medium

Source: Gustafson COLLEGE ALG page 289

74. The increase in the length of a steel bar due to a temperature increase varies directly as the product of its original length and the change in temperature. A steel girder 200 m long increases in length by 0.12 m when the temperature changes from 0C to 50C. Find the maximum temperature change possible if a 50 m steel girder can expand by no more than 0.04m.

Technology Construction 3 Variables Second Degree Medium

Source: adapted from Washington, page 170

75. When water is pumped, the engine pressure reflects the pressure of the water at the pump, and the nozzle pressure is the pressure at the nozzle; the nozzle pressure is lower because of friction loss. A standard hose size is 2.5 inches in diameter.

For this size, the formula FL = 2Q² + Q relates the friction loss (FL) per 100 feet in lb/in² to the flow rate (Q) measured in 100 gallon/min. At the scene of a fire, a 600-foot line of 2.5 in hose needs to supply 300 gallon/min, with a nozzle pressure of 100 lb/in². What should the engine pressure be?

Technology Fire Science Two Variables Second Degree Medium

Source: adapted from Globe MATH WORKSHOP page 90

76. Two ships leave port at the same time, one sailing west at a rate of 17 mi/hr and the other sailing south at 12 mi/hr. If t is the time (in hours) after their departure, express the distance d between the ships as a function of t.

Variation: If the radios on the ships have a range of 100 miles, how long can they stay in communication?

Technology Transportation Two Variables Second Degree Medium

Source: Swokowski, page 149

77. A road is to be built between two cities, A and B, which are on opposite sides of a river of uniform width (100 yards). City B is 8 miles east and 6 miles north of city A, and the river runs east and west at about 2 miles north of city A. Where should the bridge be located to minimize the total driving distance between the cities? (Assume that the bridge must be built directly across the river, not at an angle; that is, the bridge will be 100 yards long.)

Technology Transportation 3 Variables Second Degree Medium

Source: based on Bittinger APPLIED CALC page 203

78. The rate at which a hot object radiates heat varies directly with some power of its Kelvin temperature. (K = C + 273; Kelvin temperatures are sometimes called 'absolute temperatures'.) By experiment, you find that an electric heater radiates 5 calories per minute when it is at 300K, and 80 calories per minute when it is at 600 K. With what power of the Kelvin temperature does the heat radiation vary? Write an equation for this particular heater relating the heat radiation to the Kelvin temperature. Predict the heat radiation rate if the heater is at 800K.

Technology Construction Two Variables 3rd Degree Medium

Source: adapted from Foerster ALGEBRA & TRIG pages 215-216

79. A family wishes to make a mound for the children to play on, especially in the winter snow. For that purpose, they want the slope of the sides to be ¼, and the mound will be in the shape of a cone. Their budget allows them to buy 17 cubic yards of dirt for the cone. Find the approximate diameter of the cone they can build if the volume is estimated by V=r²h.

Technology Construction Two Variables 3rd Degree Medium

Source: Rotman

80. A 255-foot-long steel beam is anchored between two piling 50 feet above the ground. An algebraic representation for the amount of vertical deflection s caused by a 250-pound object placed d feet from the west piling is .

Use a graphing utility to graph this function, and estimate the location of the maximum deflection and the size of the maximum deflection.

Technology Construction Two Variables 3rd Degree Low

Source: Demana COLLEGE ALG page 220

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 LIFE: TRANSPORTATION and SAFETY APPLICATIONS

81. Extensive research has provided data relating the risk R (in %) of having an automobile accident to the blood alcohol level B (in %). An exponential model for the risk is:

According to this model, the risk of an accident goes up with the amount of alcohol consumed; common 'treatments' -- such as coffee, a cold shower, or fresh air -- will not lower the risk. Find the risk for a blood alcohol level of 0.10%, and find the blood alcohol level that has a risk of 50%. For your state, find the legal limit of blood alcohol, and compute the risk for that concentration.

A model that takes into account the limit of about 100% risk is:

Repeat the same exercises using this model.    Graph both models, and compare the advantages of each.

Life Safety Two Variables Exponential High

Source: adapted from Bittinger APPLIED CALC pages 270-272

82. After a person drinks an alcoholic beverage, the alcohol level L in the person's blood rises to a level and then falls. If the peak level of alcohol in the blood is 0.3 mg per milliliter of blood, the formula for the level after the peak is where t is the time in hours after the peak level was reached. If the legal driving limit of alcohol is 0.1 mg per milliliter of blood, how long will it be until a person will be able to legally drive?

Life Safety Two Variables Exponential High

Source: adapted from Farlow APPLIED MATH page 676

83. The July, 1974, issue of Popular Science magazine has an article comparing the costs of owning a car with a diesel engine with the cost of owning a car with a gasoline engine. According to this article, it costs about 2.5 cents per mile to operate a diesel car. Gasoline cars cost about 7 cents per mile to operate because they use more fuel, the fuel is more expensive, and they need more frequent tune-ups. According to Popular Science, you must pay about $2500 more to buy a diesel car. Write functions giving the cost in dollars to operate a diesel car and a gasoline car for d miles. How many miles must be driven with the diesel car before the operating costs are less than the gasoline car?

Life Cars & Boats Two Variables First Degree High

Source: adapted from Foerster ALGEBRA AND TRIG page 75)

84. Suppose towns A and B are 120 miles apart. If a car averages 60 mph from A to B and 40 mph on the way back, what is the average speed for the round trip?

Life Travel Two Variables First Degree High

Source: UCSMP Algebra pg 558

85. The visibility (V) to the horizon in miles varies directly with the square root of the altitude (A) in feet; the constant of variation is 1.22. How far can you see in clear weather when you are 10,000 feet up? How high should you be if you want to see 200 miles? A person standing on the ground says they can see for 20 miles in each direction; is this reasonable?

Life Travel Two Variables Radical High

Source: adapted from NCTM Applic of Secondary School Math. pg 116

86. Consider the force on the head when a person runs into a brick wall at a normal running speed of 7 meters/second. (This is equivalent to falling onto a concrete floor from a height of 2.5 meters.) The head has very little padding of its own, only a few millimeters of skin. Assume that the head stops in around 4 millimeters. Estimate the average velocity while the head is stopping, and then find the time it takes to stop. From this find the head's acceleration. (The acceleration can be estimated by subtracting the initial and terminal velocities, and then dividing by the time.) If the typical safe acceleration is about 500 meters/second for short periods, will this accident cause serious injury?

Life Safety 3 Variables Rational High

Source: adapted from Gustafson PHYSICS: HEALTH pages 145-147

87. The maximum weight that a board can support depends upon the kind of wood, its width, its thickness, and how long it is between supports. The formula for one kind of wood is:

    M = maximum load (lb); W = width (in),   T = thickness (in), L = length between supports (ft)

Marcia and John are trying to decide which way to put a 2 inch by 4 inch board that is 8 feet long. The 'flat' way, the thickness is 2 inches; the 'thick' way, it is 4 inches. What weight can the board support each way?

Variations: The floor boards in an older house are 0.875 inch thick and 8 inches wide; they are supported by beams that are 2 feet apart. What weight can they support?

What thickness of the same wood that is 10 feet long and 8 inches

wide would be needed to support a weight of 1000 lb?

Life Safety 3 Variables Rational High

Source: adapted from UCSMP Adv Algebra page 110

88. A microwave oven, a coffee pot, and a toaster are plugged into the wall outlets of a single circuit. They are connected in parallel with resistances of 15 ohms, 30 ohms, and 20 ohms respectively. What is the total resistance? Will a 15 amp circuit be safe if there are 110 volts? Two formulas given:

and .

Variation: If total resistance is 6, and two appliances are 10 ohms and 25 ohms, what resistance can be added?

Life Safety One Variable Rational High

Source: MTH 102 applic supp 5

89. According to one study of fuel economy, the mileage y (miles per gallon) for a small car is estimated by the equation

where v is the speed (miles per hour), given that the speed is between 0 and 70 mph. Find the speed to maximize the fuel economy.

Life Cars & Boats Two Variables Second Degree High

Source: LCC

90. The Department of Transportation has determined that the number of miles per gallon M that a certain model of automobile will get when it travels s miles per hour is:

M = -0.025s² + 1.40s + 5.5 (20s55)

At what speed will the automobile get the largest number of miles per gallon? What speed will minimize the miles per gallon? How many miles per gallon will this model get at those speeds?

Life Cars & Boats Two Variables Second Degree High

Source: Farlow APPLIED MATH page 607

91. Different materials have different resistances for the flow of electricity. Some are given below:

Copper Rod (1 cm thick, 1 m long)       0.0002 ohms
Iron Rod (1 cm thick, 1 m long)             0.0013 ohms
Wood Rod (1 cm thick, 1 m long)         100 000 000 000 ohms
Glass Rod (1 cm thick, 1 m long)           100 000 000 000 000 ohms
100-watt, 110-volt light bulb                  120 ohms
Electric Coffee pot                                 15 ohms
Human --
         with wet skin                                 1000 ohms
         with dry skin                                 100000 ohms

How much current would flow through a person with dry skin if he or she had a 1000-volt difference between the hands and feet? How much current would flow through a person with wet skin when the voltage was 110 volts? NOTE: V = IR, where V is the voltage, I is current (amps), and R is the resistance. Will either accident likely result in death, if 0.1 amps will cause the heart to stop beating?

Life Safety 3 Variables Second Degree High

Source: based on Gustafson PHYSICS: HEALTH pages 324-325 and 359

92. Braking distance is approximated by d = v + v² under certain conditions. If the driver decides to stop 120 feet from stop sign, how fast can car be going to stop in time?

Life Safety Two Variables Second Degree High

Source: MTH 102 applic supp 4

93. The maximum speed v, in miles per hour, at which a car can safely travel around a circular turn of radius r, in feet, is given by r = 0.42v². Plot r as a function of v.

Variation: Estimate the maximum safe speed for a turn with a radius of 300 feet. Estimate the smallest circular turn that can be done at a speed of 50 miles per hour.

Life Safety Two Variables Second Degree High

Source: Washington BASIC TECH MATH page 42

94. On a two lane road, a car is going at a constant speed of 70 feet per second when the driver decides to pass a truck. The rate of acceleration is 5 feet/second²; that is, after one second, the speed has increased to 75 feet/second. How fast, in miles per hour, is the car going after 4 seconds? If the distance travelled is given by s = 70t + 2.5t², how far has the car gone? If the safe passing zone is 600 feet, how many seconds can the driver take to pass the truck? How fast will the car be going?

Life Safety Two Variables Second Degree High

Source: Rotman

95. Your car's stopping distance is the sum of the braking distance and the reaction distance. (The reaction distance is the distance the car travels between the time you realize you need to stop and the time you put your foot on the brake pedal. The braking distance is the distance traveled after your foot starts pushing on the brake pedal.) The braking distance varies with the square of the car's speed, and the reaction distance varies directly (with the first power) with the car's speed. At 30 kph, your reaction time is 7 meters and the braking distance is 6 meters. Write expressions for the reaction distance and braking distance, and an equation for the stopping distance. Use this equation to predict the stopping distance if you drive 100 kph.

Life Safety 3 Variables Second Degree High

Source: adapted from Foerster ALGEBRA & TRIG page 217

96. Oliver Sudden is driving along a straight, level highway at 64 mph when his car runs out of gas. As he slows down, his speed decreases exponentially with the number of seconds since he ran out of gas, dropping to 48 mph after 10 seconds. Find the equation relating speed to time, and predict Oliver's speed after 25 seconds. When his speed reaches 1 mph, the car will stop moving; how many second after he ran out of gas will that happen?

Note: This model predicts Oliver would travel about 2.55 miles before stopping; this is not very reasonable.

Life Cars & Boats Two Variables Exponential Medium

Source: adapted from Foerster ALGEBRA & TRIG pages 165-166

97. Based on statements from General Motors and some estimates, the fuel usage for a compact car in city driving can be estimated by the formula F = 0.25 - 0.0025r, where F is in liters per kilometer and r is the average speed, including idling time, in km/h. Find the fuel usage when the average speed is 30 km/h. Find the average speed when the fuel usage is 0.18 L/km.

Life Cars & Boats Two Variables First Degree Medium

Source: adapted from Bolker USING ALG page 84

98. A charter boat charges $25.00 per person plus usage fee of $16.00 per hour. How many hours can 2 men rent boat if total expenses must not exceed $100?

Life Cars & Boats One Variable First Degree Medium

Source: Rotman

99. How much water should be added to 10 liters of antifreeze to get a solution that is 60% antifreeze?

Life Cars & Boats Two Variables First Degree Medium

Source: Rotman

100. An airplane flying with the wind can cover a certain distance in 2 hours. The return trip against the wind takes 2½ hours. How fast is the plane and what is the speed of the air current if the one-way distance is 600 miles?

Life Travel Two Variables First Degree Medium

Source: McKeague Interm pg 495

101. A rope that has a safe working load of 1000 newtons is stretched between two tall buildings. A tightrope walker, who has a mass of 75 kilograms (and thus a weight of 75 kg 9.8 m/s² = 735 newtons), intends to walk across it. The slack in the rope has been adjusted so that when the tightrope walker is in the middle, the rope will be depressed at an angle of 15 at each building. Is the rope strong enough to support the walker? To find the weight that the rope can support, you can use

F = 2C sin x, where F is the weight, C is the capacity of the rope, and x is the angle of depression.

NOTE: Without trig functions, sin x can be approximated by the expression  -.00011x² + .019x  (where x is between 0 and 45).

EXTENSION: C here is really a force along one side of the rope. In some cases, you can use this relationship to generate a needed force, such as for pulling a car out of mud or snow. If a strong rope is stretched VERY tightly between the car and a tree, a sideways push (F) in the middle of the rope will generate a large force (C).

For example, if we need 5000 newtons (a typical value) to pull a car out, we might use a force C of 500 newtons using an angle of about 6. This obviously requires a very strong rope (or chain) being stretched extremely tight.

Life Safety 3 Variables other Medium

Source: adapted from Gustafson PHYSICS: HEALTH page 51

102. Suppose that x fluid ounces of pure antifreeze is added to 200 fluid ounces of a 35% antifreeze solution. Write a function relating the concentration of the new mixture to x. Graph this function. What amounts of pure antifreeze would result in a mixture that is at least 50% antifreeze?

Life Cars & Boats Two Variables Rational Medium

Source: based on Demana COLLEGE ALG page 229

103. The maximum force a road can exert on the tires of a 1500 kg car is 900 newtons. (One newton is 1 kg-m/s².) What is the maximum velocity at which the car can round a turn of radius 100 meters? Since this speed is in m/s, convert it to miles/hour Note that the 'centripetal force', which keeps the car from skidding, is given by the formula: , where m is the mass, v is velocity, and r is radius.

Life Safety 3 Variables Rational Medium

Source: adapted from Beiser APPLIED PHYSICS page 69

104. On a trip to St. Louis you visit the Gateway Arch. Since you have plenty of time on your hands, you decide to estimate its altitude. You set up a Cartesian coordinate system with one end of the Arch at the origin. By stepping off, you estimate that the other end of the Arch is at x = 162 meters. To find a third point, you measure a height (y) of 4.55 meters when at x = 1 meter. You assume that the underside of the Arch is parabolic. Find the equation for the underside of the Arch, and find the vertex. How high is the peak of the Arch?

Life Travel Two Variables Second Degree Medium

Source: adapted from Foerster ALGEBRA & TRIG pages 122-123

105. Two people are in a rowboat 800 feet from the nearest point on a straight shoreline. One of the people has a heart attack, but the other one is in very good condition. She needs to get help fast, and there is a building and telephone about 2000 feet further down the shoreline. She knows that time is critical, and that she can row the boat at about 6 mph and can run at about 10 mph. How far down the shoreline should she land the boat to minimize the total time for rowing and running? (Hint: After you represent the distances on the shoreline, express the time needed for rowing the 'hypotenuse' and running the shoreline as fractions. Then write a single function for total time, & graph this function.)

Extension: What ratio of boat and running speeds would mean the person should row directly to the nearest point on the shoreline?

Life Safety Two Variables Rational Medium

Source: adapted from Swokowski CALCULUS page 212

106. At an altitude of h feet above sea level, the boiling point of water is lower by a certain number of degrees than the boiling point at sea level, which is 212F. The difference is given by the approximate equation T² + 520T - h = 0. An explorer is camped at the 10,000 foot level on a mountain. Since fires are hard to maintain, she estimates that she can only heat water to about 200F. Will she be able to boil the water?

Life Safety Two Variables Second Degree Medium

Source: adapted! from Washington, page 328

107. The table below gives the number of vehicles involved in accidents per 100 million miles of travel (for nighttime driving).

Use the values for 20 mph, 50 mph, and 80 mph to find a quadratic equation that fits the data. Use this model to predict the number of vehicles involved in accidents at 30 mph, and compare this to the known value in the table. Also, predict the number of vehicles involved in accidents at 25 mph and 35 mph. Why are there fewer accidents at speeds of 50 to 60 mph?

Life Safety Two Variables Second Degree Medium

Source: from Bittinger APPLIED CALC

108. The maximum extension for an extension ladder is 22 feet. The safety guidelines on the ladder state that the distance from the base of the ladder to the wall must be one-fourth of the height of the top of the ladder. How high on the house can the top of the ladder be?

Life Safety One Variable Second Degree Medium

Source: Rotman

109. A 2-kilowatt water heater is to be connected to a 240 volt power line whose circuit breaker is rated at 10 amps. Will the breaker 'open' (or trip) when the heater is switched on? Note that P = IV, where P is the power (watts), I is current (amps), and V is the number of volts. A kilowatt is 1000 watts.

Life Safety 3 Variables Second Degree Medium

Source: from Beiser APPLIED PHYSICS page 207

110. The breaking strength of a given type of rope is proportional to the cross-sectional areas. A braided nylon rope 1 inch in diameter has a breaking strength of 28,500 pounds. Find the breaking strength of similar ropes that are 0.5 inch and 2 inch in diameter.

Life Safety Two Variables Second Degree Medium

Source: from Beiser APPLIED PHYSICS page 118

111. Suppose a 1200-watt electric heater and a large, 1000-watt light are plugged into the same 110-volt circuit. Will the 15-amp circuit breaker break if they are both turned on at the same time? Note that p = IV, where p is the power (watts), I is the current (amps) and V is the voltage.

Life Safety 3 Variables Second Degree Medium

Source: Gustafson PHYSICS: HEALTH page 319

112. When two masses of water are mixed, the temperatures and masses are related by the equation m₁(T₁ - T) =

m₂(T - T₂), where m₁ and T₁ are the mass and temperature of the hotter water, and T is the final temperature of the mixture. A family's hot water heater is set at 130F, and the cold water is 55F. If 5 kg of hot water is mixed with 3 kg of cold water, what will be the final temperature of the mixture? If 1 kg of hot water is used, how much cold water should be added to have a mixture that is 100F?

Life Safety 3 Variables Second Degree Medium

Source: based on Aufmann INTRO ALG page 112

113. The stopping distance S for a car after the brakes are applied is directly proportional to the weight W and the square of the speed R. A car weighing 1800 pounds and traveling 40 miles per hour will stop in 86.4 feet. Determine the equation for S, and use it to find the stopping distance for a car weighing 2400 pounds and traveling 60 miles per hour.

Life Cars and Boats 3 Variables Degree Medium

Source: Varberg APPLIED CALC page 318

114. Scuba diving equipment indicates it is save to a pressure of 3.6 atmospheres. If P = 1 + D/33, what depth can she dive to?

Life Cars & Boats One Variable First Degree Low

Source: MTH 102 applic supp 5

115. The stopping on glare ice (at some fixed speed) of regular tires is given by a linear function of the air temperature F: D = 2F + 115, where D is the stopping distance in feet on ice, when the air temperature is F (F). Find the stopping distance when the temperature is 0F, -20F, and 32. Is ice more slippery when it is warmer?

Life Safety Two Variables First Degree Low

Source: Bittinger APPLIED CALCULUS page 48

116. Two cars pass the same point at the same time. One car is going at a constant speed of 80 feet per second. The second car has been accelerating at a rate of 5 feet/second² and its speed at that point is 70 feet per second. Find the time(s) that either car was or will be 200 feet from the point where they were together. (For the accelerating car, its distance travelled is given by s = 70t + 2.5t².)

Life Safety Two Variables Second Degree Low

Source: Rotman

117. Ignoring friction, the theoretical mechanical advantage of a ramp is the ratio of the length to the height. Friction reduces this advantage, and this is reflected in an 'efficiency factor". A ramp 80 feet long slopes down 5 feet to the edge of a lake. How much force is needed to pull out an 800-pound boat by using a 200-pound trailer if friction reduces the efficiency to 90%? (For any machine, the force needed is found by dividing the weight being moved by the advantage after efficiency is taken into account.)

Life Cars & Boats Two Variables Rational Low

Source: from Beiser APPLIED PHYSICS page 112

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LIFE:  OTHER APPLICATIONS

118. Dr. Dickler, in a high school, asked his class this question:

How much would a horse weigh standing on two legs if it weighed two thousand pounds when standing on four legs?

Life Other One Variable First Degree high

Source: adapted from Bolker USING ALG pg 64

119. Monthly payments for mortgages can be calculated with: where P is the Principal, r = rate; t = years. What is the monthly payment on a $45000 mortgage for 30 years if the rate is 10%? If a family needs to have a payment of less than $500 per month, what mortgage amount can they afford if the interest rate is 9% and the mortgage is for 30 years?

Life Consumer 3 Variables Exponential High

Source: Rotman

120. If you invest P dollars in an account with an annual interest rate r that is compounded n times per year, then t years later the amount of money in the account will be:

If $5000 is placed in an account with an annual interest rate of 12% compounded twice a year, how much money will be in the account 10 years later?

Variation: How long does it take for $5000 to double if it is deposited in an account that pays 5% annual interest compounded once a year?

Life Consumer 3 Variables Exponential High

Source: McKeague Interm pg 616)

121. The number of minutes it takes to cook bacon in a microwave oven depends on how many slices you put in at once. A popular brand of oven specifies 1.75 minutes for 2 slices, and 2.5 minutes for 4 slices. Assume that the cooking time varies directly with some power (not necessarily an integer) of the number of slices. Use the two given ordered pairs to derive an equation relating the cooking time to the number of slices. How long would it take to cook 8 slices? 6 slices? 1 slice?

Life Cooking Two Variables Exponential High

Source: adapted from Foerster ALGEBRA & TRIG page 237

122. The number of hours (h) that milk stays fresh is a function of the surrounding temperature (t). Use the formula:

where t is degrees F.

How long newly pasteurized milk will stay fresh when stored at temperature is 46 F?

Life Cooking Two Variables Exponential High

Source: UCSMP Adv Algebra page 460

123. The boss says: "Hard times have come. Everyone takes a 10% pay cut." Then he says: "Prosperity returns. Everyone gets a 10% pay raise. Now we are all even, okay?" Is it okay? What if the raise preceded the cut?

Life Consumer One Variable First Degree High

Source: Bolker USING ALG page 124

124. Newton's Law of Cooling relates the temperature of a hot object as it cools based on the time. The general form is:

where T_e is the temperature of the surrounding environment, T₀ is the initial temperature, and t is the time in minutes.

The constant k reflects the rate of heat exchange, and depends on a number of factors; a typical value of k would be 0.028. If such an object is removed from an oven when its temperature is 350F into a 70F room, find its temperature after 45 minutes. If the recipe says it should cool completely before the next step (frosting), how long must we wait before the temperature is 90F or lower?

Life Cooking 3 Variables Exponential High

Source: adapted from Larson ALG & TRIG page 365

125. The light intensity at a depth of x meters under water is related to the light intensity at the surface by the model:

where I₀ is the intensity at the surface, and k depends on the murkiness of the water.

This is called the Bougour-Lambert Law. Suppose that the constant k has been determined experimentally to be 0.69 for a certain region of a lake. How much of the light intensity at the surface reaches a depth of 2.5 meters? A k value of .69 represents very murky water. If the water was much clearer, the k value might be .1; how much of the light intensity at the surface would reach the 2.5 meter level?

Life Sports Two Variables Exponential High

Source: adapted from Farlow APPLIED MATH pages 658-659

126. The rate for personal ads in a newspaper were as follows:

3 lines for $10.20
4 lines for $13.60
5 lines for $17.00.

What would you expect to pay for a seven line message? What is the lowest possible price for an ad, if the customary policy of "a fraction of a line is charged as a full line" is followed?

Life Consumer Two Variables First Degree High

Source: adapted from Bolker USING ALG page 86

127. You are considering purchasing one of two cars, both four years old. One car costs $3000 and gets 20 miles per gallon. The other costs $4500 and gets 45 miles per gallon. The maintenance of the cheaper car is expected to be $400 per year plus $.08 per mile. The maintenance of the 2nd car is expected to be $300 per year plus $.06 per mile. The cost of gasoline is expected to remain at about $1.15 per gallon for the next 36 months. If you average 10,000 miles per year, how long would you need to own the car before the second would have less total costs?

Life Consumer Two Variables First Degree High

Source: Rotman

128. The bank service charge for a checking account is $1.12 per month, plus $.08 per check; how many checks can be written for service charge to be not more than $3.50 per month?

Life Consumer One Variable First Degree High

Source: MTH 102 applic supp 1

129. Data-M company pays salary of $150 per wk plus 4% commission. TrueTech pays only 6% commission. What sales give a higher salary at TrueTech?

Life Consumer One Variable First Degree High

Source: MTH 102 applic supp 1

130. For phone marketing, one company pays its callers flat rate of $7.49 per hour, while a second company pays $6.10 per hour plus $.40 per appointment made with a customer. How many appointments would have to be made with second company per hour to earn more per hour?

Life Consumer One Variable First Degree High

Source: MTH 102 applic supp 1

131. The charges for a certain long distance phone call are $1.43 for the first 3 minutes, then $.28 for each additional minute or portion of a minute. How many minutes can she talk if she doesn't want to spend more than $5.00?

Life Consumer One Variable First Degree High

Source: MTH 102 applic supp 1

132. A customer and a store are in an argument. The store computed the bill by adding the sales tax and then taking a discount of 10%. The customer was sure she could get a better deal if the discount was taken first, the then add the sales tax. Who is right?

Life Consumer One Variable First Degree High

Source: Rotman

133. Charlotte now earns $6 per hour as a grounds keeper working 35 hours per week. She has been offered a similar job paying $280 per week. How much of an increase in her hourly wage must her present employer give Charlotte to meet the better offer? What

percent increase is this?

Life Consumer One Variable First Degree High

Source: Pulsinelli/Hooper Intro pg 205

134. A recipe calls for 1 cup of 2% milk. However, we have only whole milk and skim milk, which have 3.6% and 0% milkfat respectively. How much of each should we mix to get 1 cup of 2% milk?

Life Cooking Two Variables First Degree High

Source: Rotman

135. The cost of heating a house in a region with cold winters depends linearly on the number of degree days. When the number of degree days is 4000 the total cost for heating is $1050. When there are 5600 degree days the cost is $1425. Find the equation relating the cost to the number of degree days. What would it cost to heat the house in a winter in which there were 5300 degree days? (note: model is C = 112.5 + .234D)

Variation: If the family has $1600 available for heating costs, what is the maximum number of degree days in the winter before they run out of this fund?

Life Cooling & Heating Two Variables First Degree High

Source: Bolker USING ALG page 30

136. A certain lawn measures approximately 30 feet by 40 feet. In the past two different mowing techniques have been used. First, one mowed lengthwise, made a 180 degree turn, again mowed lengthwise, made a 180 degree turn, and so on. In the second method, one went around the perimeter, starting with the full lawn and then around the smaller rectangles that resulted; this method uses 90 degree turns. If the time for a 180 degree turn is twice the time for a 90 degree turn, which method should be used to save time?

Life Home Maintenance Two Variables First Degree High

Source: Sloyer FANTASTIKS page 21

137. Assume that you are 40 feet above the baseball field, and a ball is hit straight up at a speed of 88 feet/second. Use h = 88t - 16t², and v = 88 - 32t.

A certain film type can take good pictures as long as the speed of an object is less than 70 feet per second. Could you use this film to take the picture when the ball is even with you? How many seconds will you have to take a picture at any height?

Life Sports Two Variables Second Degree High

Source: Rotman

138. What velocity must a ball have when thrown upward if it is to reach 100 ft? Note that h = vt - 16t² describes the height of the ball on the way up and on the way down, and that h = 16t² represents how far the ball falls after reaching the maximum height.

Life Sports 3 Variables Second Degree High

Source: from Beiser APPLIED PHYSICS page 31

139. Megan and her mother are flying a kite. She has unrolled 30m of string, and the kite is 25m from her along the ground. How high is the kite?

Life Sports One Variable Second Degree High

Source: UCSMP Algebra pg 361

140. A man and his daughter are playing with a baseball and wonder how high they can throw the ball. Carefully, they find that the ball is in the air for 6 seconds. How high can they throw the ball? Reference: h = vt - 16t² and V = v - 32t.

Life Sports 3 Variables Second Degree High

Source: Rotman

141. In a baseball stadium, in the upper deck, you are 40 feet above playing field. A batter hits the ball straight up at speed of 88 feet per second. Using

h = vt - 16t² and V = v - 32t, at what time(s) will the ball be exactly at your height? How fast will it be traveling?

Life Sports Two Variables Second Degree High

Source: adapted from MTH 102 applic supp 4

142. Suppose that you are shopping at a supermarket and find that the Texas grapefruits have twice the diameter of the Florida ones, but cost seven times as much. Recalling that the volumes of similarly shaped solids vary directly with the cube of a linear measurement such as a diameter, you instantly determine which kind of grapefruit gives you more for your money. Which one does, and why?

Life Consumer One Variable 3rd Degree High

Source: Foerster ALGEBRA & TRIG page 217

143. The U.S. Postal Service will accept a package for parcel post only if its length plus girth (shortest distance around the package) does not exceed 108 inches. Mail-It-Secure plans to market a box that will satisfy this condition and have maximum volume. If the box is to have square ends and rectangular sides, write an equation showing volume in terms of x, the length of a side of the square end. Graph this function, and from the graph, find the value of x that gives the maximum volume.

Life Consumer Two Variables 3rd Degree High

Source: adapted from Farlow APPLIED MATH page 618

144. Consider a $80,000 house, an inflation rate of 5%, and a depreciation rate of 3% of the original value per year. What will the value of the house be in 10 years? When will the house be worth $200,000?

Life Consumer Two Variables Exponential Medium

Source: adapted from NCTM Applic of Secondary Math pg 141

145. Assume that the number of hours milk stays fresh decreases exponentially with temperature. Suppose that milk in the refrigerator at 0C will keep for 192 hours, and milk left out in the kitchen at 20C will keep for only 48 hours. Find an equation relating keeping time (hours) to temperature (C). Find the ratio of the keeping time at 40C compared to 20C.

Life Cooking Two Variables Exponential Medium

Source: adapted from Foerster ALGEBRA & TRIG page 166

146. The time it takes for milk to sour depends on the temperature at which it is stored. An exponential model of this relationship has the form: , where T is the number of days, F is the temperature (F), and A and k are constants. Here is some data:

Find the constants A and k. Then estimate the life of milk stored at 36F and at 90F.

(model is: -- or some approximation to this)

Life Cooking Two Variables Exponential Medium

Source: adapted from Bolker USING ALG page 201 and 43

147. The temperature of a cup of hot coffee was 180F, but after 10 minutes the temperature was 170F. If the surrounding temperature is 60F, use the general Newton's Law of Cooling to write an equation to describe the general cooling pattern. How long will it take to reach the lowest temperature most coffee drinkers can enjoy (about 120F)?

Life Cooking 3 Variables Exponential Medium

Source: adapted from Farlow APPLIED MATH page 661

148. Consider a cup of tea heated to 210F. The room temperature is 70F. The exponential function: , where y is the temperature after t minutes. Since the tea should steep for 10 minutes before serving, it will be cooler when it is drunk. What will the temperature be after it steeps for 10 minutes? If a child is going to sip the tea, it should be no hotter than 130F; how many minutes would it take to reach this temperature?

Life Cooking 2 Variables Exponential Medium

Source: based on Piascik COLLEGE MATH page 136

149. The world records for running races are often modeled by linear functions. For example, the mile run record can be predicted by the equation R = -0.329x + 264.5, where R is in minutes, and x is the years since 1875. However, this model predicts that the record can fall to any level, including a 1-minute mile; the 1-minute mile would be run in the year 2465. The exponential model does a much better job; one such model is . Compare the predicted world records with both models for the years 1980, 2000, and 2200.

Note: A limited growth model would be better yet, if someone would generate it.

Life Sports Two Variables Exponential Medium

Source: based on Bittinger APPLIED CALC pages 66 and 463

150. The faster an object falls, the greater the air friction. Eventually, as the speed increases, air friction will increase to the point where its force upward is equal to the weight of the falling object. At that point, the object has reached its terminal velocity. Some typical terminal velocities are:

For irregular shapes, such as stones, these values can vary greatly.

The speed of a baseball in free fall after t seconds can be estimated by the formula , and the distance fallen by the formula . A baseball is hit so that it reaches a height of 50 meters before it starts to fall; find the speed and distance travelled 3 seconds after it starts to fall.

Life Sports 2 Variables Exponential Medium

Source: terminal velocities from Gustafson PHYSICS: HEALTH page 160

151. Bank A charges $1.55 per month plus $.12 per check; Bank B charges only $1.00 per month but has a per check fee is $.15. After how many checks will Bank A become less expensive than Bank B?

Life Consumer One Variable First Degree Medium

Source: MTH 102 applic supp 1

152. Gary has $1760 in his savings now. If he deposits $40 per week in this account, for how many weeks must he continue to save if he wants a total of $5000 in his account?

NOTE: good variation -- same setup but consider the effect of interest. (Would have to say that he deposits the money at the end of each week and specify a 5% annual interest rate compounded monthly: without interest: 81 weeks, the average balance per month is about $3400; interest would be about $290--which is about 7 weeks of savings. Estimated answer: 74 weeks.)

Actual answer: 75 weeks

for original balance; n = weeks/4 (months);

for remainder (i=.05/12); solve for n given R and F.

Life Consumer One Variable First Degree Medium

Source: Pulsinelli/Hooper Intro pg 203 for original; see also Farlow APPLIED MATH page 246)

153. A person currently weighs 97 lb, and is gaining ½ lb per week; how many weeks will it take before the weight is at least 110 lb?

Life Health One Variable First Degree Medium

Source: MTH 102 applic supp 1

154. The RDA for ascorbic acid is 45 mg, and 14 mg for niacin (for women between ages of 19 and 22). Each ounce of Cereal A has 10 mg of ascorbic acid and 4 mg of niacin. Cereal B has 15 mg of ascorbic acid and 2 mg of niacin per ounce. If she eats a mixture of the two cereals, how much of each should she eat to get the RDA for both ascorbic acid and niacin?

Life Health Two Variables First Degree Medium

Source: based on McKeague Interm pg 502

155. A bullet is fired horizontally at a target, and the sound of its impact is heard 1.5 seconds later. If the speed of the bullet is 3300 ft/sec and the speed of sound is 1100 ft/sec, how far away is the target?

Life Sports (hunting) One Variable First Degree Medium

Source: Swokowski, page 74

156. The pressure under h feet of a liquid is given by: , where p_atm is atmospheric pressure (14.7 lb/in²) and d_w is the density by weight (lb/in³) of the liquid. What is the pressure, in lb/in² at the bottom of a swimming pool 6 feet deep that is filled with fresh water? (Water density is 62 lb/ft³.)

Life Sports Two Variables First Degree Medium

Source: from Beiser APPLIED PHYSICS page 143

157. A family has a portable circular swimming pool for the children, which measures 8 feet in diameter and can hold 2 feet of water. They are going to fill it with their garden hose, which has a ½ inch diameter nozzle opening. How long will it take to fill the pool? You are given:

; ; 1 cubic foot is about7.5 gallons of water

where Q = flow rate (gallons per minute)

D = diameter of nozzle      P = water pressure, which is about 10 lb/in²

Life Sports 3 Variables Rational Medium

Source: adapted from Saunders WHEN ARE WE EVER GONNA page 104

158. The instruction booklet for a video cassette recorder (VCR) includes a table relating the counter readings and the time the tape has run.

Use the first three data points to find a quadratic equation that fits the data. Using this equation, predict the counter readings for 1.5 hr and 2.5 hr; also, if the counter reads 564, how long has the tape been running?

Life Consumer Two Variables Second Degree Medium

Source: from Bittinger APPLIED CALC page 72

159. How much energy in kilowatthours does a 240 volt clothes dryer that draws 15 amps use in 45 minutes of operation?

Life Consumer 3 Variables Second Degree Medium

Source: Beiser APPLIED PHYSICS page 206

160. Martha has a square garden measuring n feet on each side. She will enlarge it so that each side is 1½ times as long as it was. Write an expression for the area of the enlarged garden. Write the increased area as a percent of the original.

Life Home Maintenance One Variable Second Degree Medium

Source: adapted from Pulsinelli/Hooper Intro page 243

161. A rock is dropped into a well. Three seconds later, the sound of the splash is heard at the top of the well. How deep is the well?

Life Other Two Variables Second Degree Medium

Source: Hestenes ALG AND TRIG page 75

162. In round-robin tournament, number of games given by g = ½n(n - 1) where n is the number of teams. Rented a playing field for one week, and 4 games can be played each day. How many teams can be invited?

Life Sports Two Variables Second Degree Medium

Source: MTH 102 applic supp 4

163. Two baseball players are arguing about when a ball is going faster. One says that when the ball is thrown horizontally and caught it is going faster, while the other says that when it is thrown vertically and caught it is going faster. The first argues that gravity will not slow it down, and the second says that gravity will speed it up on the way down. Assume that they can each throw the ball at a speed of 80 feet/second in either direction. Who is right? Reference: h=80t-16t², and v=80-32t; ignore the effects of friction.

Variation: Other sports, such as football, where the speed of the ball when caught could affect a person's ability to catch it.

Life Sports 3 Variables Second Degree Medium

Source: Rotman

164. You learn that a baseball "diamond" is really a square with sides that are 90 feet long. 1st base and 3rd base are at opposite corners; how far apart are they?

Variation: Home base and 2nd base are at opposite corners. When a runner tries to "steal" 2nd, the catcher throws the ball after the pitcher throws the ball home. The pitcher is located about 60 feet from home base. How long does it take for the ball to get from the pitcher to 2nd base (by way of the catcher) if the catcher and pitcher both throw at 100 ft/sec?

Life Sports Two Variables Second Degree Medium

Source: Rotman

165. A hot air balloon goes 500 feet straight up from park; wind then blows the balloon 1200 feet east. How long will it take a sound to travel from Mark (on take off spot) to Tonya (in balloon), if sound travels at a rate of about 1100 feet per second?

Variations: Two police cars leaving intersection, one going 300 yards due north and the other traveling 600 yards due east. How long will it take a sound to travel from one car to the other (assuming the sound is not transmitted by radio)?

Life Sports One Variable Second Degree Medium

Source: based on MTH 102 applic supp 4

166. When a substance melts, heat needs to be added. This quantity is called the heat of fusion, which for water is 80 kcal/kg or 144 Btu/lb. When a substance changes to a gas, the heat that is needed is called the heat of vaporization; for water, this is 540 kcal/kg or 972 Btu/lb. Find the minimum amount of ice at -10C needed to bring the temperature of 500 g of water at 20C down to 0C.

Life Cooking 3 Variables 3rd Degree Medium

Source: Beiser APPLIED PHYSICS page 161

167. In preparing tea, 600 g of water at 90C is poured into a 200 g china tea pot which has a temperature of 20C. What is the final temperature of the water? Reference: Q = mcT; some specific heat capacities are:

Life Cooking 3 Variables 3rd Degree Medium

Source: Beiser APPLIED PHYSICS page 160

168. A cook has two baking pans available for a cake recipe. One pan is rectangular, 9 inches by 13 inches, while the other is a square pan that is 10 inches on a side. If the cake will be 2 inches thick in the square pan, how thick will it be in the rectangular pan?

Variation: A box pizza has directions "Spread dough to edges of a 10 inch by 14 inch cookie sheet." How big of a circular pizza could you make with this dough, assuming it is spread the same thickness as for the rectangular pizza?

Life Cooking One Variable 3rd Degree Medium

Source: adapted from UCSMP Adv Algebra page 312

169. Different substances respond differently to the addition or removal of heat; this is reflected in the specific heat capacity. Some specific heat capacities are:

The heat gain or loss is given by Q = mcT, where m is mass, T is the temperature change, and c is the specific heat capacities. Three pounds of water at 100F are added to 5 pounds of water at 40F. What is the final temperature of the mixture? (The heat loss is equal to the heat gain.)

Life Cooling & Heating 3 Variables 3rd Degree Medium

Source: adapted from Beiser APPLIED PHYSICS pages 158-160

170. Race car drivers and designers must concern themselves with air resistance. Some of the power generated by the engine must be used to overcome this resistance. A formula can be used to estimate the horsepower required to overcome air resistance for a car with a given frontal area, as follows:

where P is horsepower, s is the car's speed in miles per hour, and A is the car's frontal area in square feet. Evaluate the power required to overcome air resistance for a car that has a frontal area of 40 square feet, traveling at a speed of 65 mph and at 130 mph. Does twice the speed require twice the horsepower to overcome air resistance?

Also, estimate the maximum speed for a car with 500 horsepower available if the frontal area is 30 square feet.

Life Sports 3 Variables 3rd Degree Medium

Source: based on CORD APPLIED MATH

171. When a boat is going at high speed, most of the power generated by its engine goes into forming the wake. The amount of power used to generate this wake is directly proportional to the seventh power of the speed of the boat. Suppose that a boat going 10 knots (nautical miles per hour) uses 0.1 horsepower for wake generation. Write the equation relating power for the wake to speed. How much power goes into wake generation when this boat goes 20 knots? 30 knots?

Life Sports Two Variables 3rd Degree Medium

Source: from Foerster ALGEBRA & TRIG pages 213-214

172. The horsepower required to propel a motorboat is proportional to the cube of the speed of the boat. Suppose that 4.00 hp is needed to drive a boat at 8.00 mi/hr. What power is required to drive it at 12.0 mi/hr?

Life Sports Two Variables 3rd Degree Medium

Source: adapted from Washington, page 170

173. The speed of sound in air at 0C (or 273K -- Kelvin) is 1087 ft/sec, but this speed increases as the temperature rises. If T is temperature in K, the speed of sound v at this temperature is given by . Compare the speeds of sound when the temperature is -10C and 30C. Would a sound speed of 1200 ft/sec be reasonable for any location on earth?

Life Other Two Variables Exponential Low

Source: based on Swokowski CALCULUS page 161

174. Given: (T, t = # teeth; R, r = revolutions per minute). A 10-speed bike in lowest gear has 39-tooth gear attached to pedal and a 24-tooth gear attached to back wheel. If Bill is pedaling at 60 rpm in lowest gear, how fast is he moving?

Life Sports 3 Variables Rational Low

Source: MTH 102 Applic Supp 5

175. A home carpenter is applying a force of 400 newtons to the end of a hammer, 28 cm from a board, in attempting to pull out a nail. The nail is 7 cm from the end of the hammer where the pivot is taking place. How much force is acting on the nail? This problem deals with equal torques, like a teeter totter; this formula applies: where the F's are the forces on the hammer and nail, and the L's are the lengths (distances) from the pivot.

Life Home Maintenance Two Variables Second Degree Low

Source: adapted from Gustafson PHYSICS: HEALTH page 79

176. A farmer wants to fence a rectangular garden. One side of the garden will be against the side of her house, and the other sides must be fenced with 100 feet of fence. Find the dimensions of the garden with the largest possible area.

Life Home Maintenance Two Variables Second Degree Low

Source: Demena/Leitzel

177. The Earth's mean radius is 6371 km. How far is the horizon of the Earth from a hot air balloon whose altitude is 2056 meters? Note that the line from the balloon to the horizon forms a right angle with the Earth's radius.

Life Sports One Variable Second Degree Low

Source: adapted from Barros-Neto, COLLEGE ALGEBRA page 117

178. When a substance melts, heat needs to be added. This quantity is called the heat of fusion, which for water is 80 kcal/kg or 144 Btu/lb. When a substance changes to a gas, the heat that is needed is called the heat of vaporization; for water, this is 540 kcal/kg or 972 Btu/lb. Five kg of water at 40C is poured on a large block of ice at 0C. How much ice melts? (Note that the heat loss equals the heat gain; for the water, Q = mcT; for ice, Q = mL where L is the heat of vaporization.)

Life Cooking 3 Variables 3rd Degree Low

Source: adapted from Beiser APPLIED PHYSICS page 158-161

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  SOCIAL SCIENCE APPLICATIONS

179. A general form of the learning curve is , where P is the performance, M is the maximum performance, B and r are constants for a given learner and a given learning situation, and t is the time. The formula reflects the fact that learning is more rapid in the initial periods, and the rate of learning slows as the performance approaches the maximum. In one situation, an average new employee is initially tested and was found to have a score of 31 out of a maximum of 100. Following two weeks of training, her score improved to 53. Use these values to find the equation of this learning curve relating P to the time in days. Also, find the time needed to earn a score of 80. Compare the time needed to achieve scores of 90 and 95. Do you see why learning to perfection is not always a good goal?

Social Science Education Two Variables Exponential High

Source: LCC

180. Suppose that the population in a small city is 32,000 in the beginning of 1990 and that the city council assumes that the population size t years later can be estimated by the equation . Approximately when will the city have a population of 50,000?

Social Science Government 3 Variables Exponential High

Source: McKeague Interm pg 618

181. A town with population 67,000 is losing 5% of its population each year. At this rate, how many people will be left in the town after 10 years?

Variations: To support basic city services through taxes, the town needs to have at least 30,000 residents. How long until there are not enough residents to support basic services?

It is estimated that 25% of the population at any one time are school age. How many students could be in the city schools after 5 years?

Social Science Government Two Variables Exponential High

Source: UCSMP Algebra pg 434

182. While common sense suggests that the electoral college method of electing presidents gives more power per voter to states with small populations, an analysis of the swing power of the states with larger populations has led some political scientists to the opposite conclusion. They propose the "three-halves rule" according to which the effective voting strength of a state in a presidential election is proportional to its population to the 1.5 power. Based on this model, the amount A that a candidate spends in a state should vary with the population according to the formula

Alfonso Hornblower plans to spend $1 million in his home state with a population of 4.5 million. Use this information to determine k in the model, and then predict how much he should spend in New York state if New York has a population of 17.5 million people. If Hornblower spends money in all 50 states and the District of Columbia, find the population that he can campaign for if he has $50 million available.

Social Science Government Two Variables Exponential High

Source: adapted from Varberg APPLIED CALC page 61

183. In his math course, Peter had a 93.6 average on his unit tests, which count for 75% of his overall average. What is the lowest score he can get on the final (25% of overall average) to keep his average above 93?

Social Science Education One Variable First Degree High

Source: Rotman

184. After 18 credits, Mahesh has a GPA of 3.1. What must he average over the next 14 credits to have an overall GPA of 3.25?

Social Science Education Two Variables First Degree High

Source: adapted from UCSMP Algebra pg 564

185. A park is being planned for part of a vacant city lot, which is 250 yards long and 125 yards wide. Fencing needs to be installed around the park, with two openings 2 yards wide each. The fenced area is to be kept in the same proportions as the lot length and width. The park itself will be primarily grass, which needs to be mowed several times each year. The cost of fencing is $30 per yard, and the cost of mowing is expected to be $.10 per square yard for the first year. Graph the total cost of the park as a function of the width of the park.

Social Science Government Two Variables First Degree High

Source: Rotman

186. A school is to be built along a road between two factories (A and B) that are 8 miles apart. Both factories emit air pollutants. At a point x miles from factory A, the concentration of pollutants (p) is measured as

where c is a constant.

How far from factory A should the school be built to minimize the air pollution problem? Hint: Although the solution does not depend on the value of c, you might wish to choose a convenient value for c (such as 2), and graph the resulting function.

Social Science Education Two Variables Rational High

Source: from Varberg APPLIED CALC page 144

187. Helicopter is searching for wreckage in ocean in a rectangular region length 12 miles and width 5 miles. The helicopter starts at one corner and flies directly to the opposite corner at a constant speed of 26 mi/hr. How long before it reaches the end of the search area?

Social Science Other Two Variables Second Degree High

Source: MTH 102 applic supp 4

188. In a study of U.S. wage earners in 1985, the Department of Labor found that the lowest-paid 25% of all wage earners earned only about 10% of the total wages paid in that year. The bottom 50% of all wage earners earned 26% of the total wages. After analyzing the data, a formula can be found that estimates the portion of total wages (L) earned by the portion of wage earners (x). In this case, the formula L = x² + 0.08x reflects the data. Once this formula is known, a Coefficient of Inequality can be computed; for this formula, the Coefficient of Inequality (CI) is given by CI = 2(-.333x³ + .46x²). To find the Coefficient of Inequality, we substitute 1 for x in this formula. What is the CI for the United States based on these formulas? The result can be interpreted by comparing it to the ideal and to other countries. In the ideal, the CI would be 0; in this case, the earned wages are fairly distributed. Other countries have a CI ranging from 0.18 for Sweden to 0.22 for France, and 0.34 for Brazil.

Social Science Government 3 Variables 3rd Degree High

Source: based on Farlow APPLIED MATH pages 747-749

189. A soup company produces a line of soups which are sold in the traditional cylindrical metal cans, which vary in size. Each morning, the stamping machines are adjusted to cut the circular ends and the vertical sides of the cans for the cans to be packed that day. The technician who sets the machines is known to make errors of up to 3% in radius measure and 5% in height measure. This means that the actual radius x is related to the ideal radius r by the inequality x - r 0.03r. Write expressions for the minimum radius and height produced by the stamping machine.

Then use these expressions to write expressions for the minimum and maximum volumes. What is the error rate for the volume?

Social Science Government 3 Variables 3rd Degree High

Source: LCC

190. An instructor uses a placement test to predict students' grades. The equation is g = 0.2t - 2.1, where t is the score on the test. What is the expected grade for a student who receives a score of 15 on the test? What is the lowest score that would predict a 3.0 grade?

Social Science Education Two Variables First Degree Medium

Source: Rotman

191. Psychologists have found that when a person learns a new task, learning is rapid at first. Then, as time passes, learning tends to taper off. Once the task is mastered, the person's level of performance approaches an upper limit. The function that relates a learner's performance with the time elapsed is called a learning curve. Learning curves are often expressed by exponential functions. Consider the learning curve expressed by:

, where y is the number of items produced on the xth day following the training period.

Find the items produced during the training period (x=0), 3 days after the training period, and 2 weeks after the training period. When will the worker produce 39 items?

Social Science Education Two Variables Exponential Medium

Source: adapted from Piascik COLLEGE MATH page 135

192. When a living organism dies, the oxygen/carbon dioxide cycle common to all living things ceases and carbon-14, a radioactive isotope with a half-life of 5700 years, is no longer absorbed. By measuring the amount of carbon-14 present in an ancient object, archaeologists can estimate the object's age. How old is a wooden statue that contains only of its original carbon-14 content? (Assume that the statue was made immediately after the tree was cut down.)

Social Science Other Two Variables Exponential Medium

Source: adapted from Gustafson COLLEGE ALG page 377

193. In psychology, experiments often deal with a subject's response to a stimulus. Two models have been proposed to predict the strength of a response based on the strength of the stimulus. Both models involve exponents and a constant which varies depending on the type of stimulus (light, taste, smell, etc). Two sample models are:

Weber-Fechner Law where S₀ is a threshold stimulus

Brenano-Stevens Law

In both models, S is the stimulus measure and R is the response measure. Compute the response predicted by each model for this data:

Social Science Psychology 3 Variables Exponential Medium

Source: adapted from Bittinger APPLIED CALC pages 547-549

194. The U.S. annual divorce rate (D) is approximated by:

where t is the number of years since 1900. According to this model, how many divorces occurred in 1960? When will there be 1 million divorces in a year?

Social Science Government Two Variables Exponential Low

Source: based on Bittinger APPLIED CALC page 345

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  BIOLOGY APPLICATIONS

195. From measurements, the level of an air pollutant in Lansing is increasing at a rate of about 8% per year. In 1991, the pollutant was at 300 mg/L. Graph the level of the pollutant for the years 1985 to 2015. When will the level of the pollutant exceed the safety standard of 800 mg/L?

Biology Environment Two Variables Exponential High

Source: Rotman

196. A model for the number of a fish (or any species) is:, where B is the number of fish at time t (in years), M is the maximum number that can be supported in the local ecology, B₀ is the initial number, and r is the rate of reproduction (a decimal value between 0 and 1). This model reflects the fact that the growth in the population slows down as the number of fish approaches the limit. One example is: , which shows that 1700 is the maximum, 10% of the maximum was the starting value, and the rate of reproduction is 0.25. For this same situation, a better model is the formula: . Find the estimated fish population after 5 years with each model, and use a graphing utility to graph each model.

Variation: A lake is seeded with 2500 perch, and the population is about 3000 one year later. If the rate of reproduction is 30%, use the model to estimate the maximum number of perch that the lake can support.

Bonus points: The reproduction rate does not take the loss of perch to fishing into account. What annual maximum catch (in number of perch) should be allowed if the perch population is to be kept within 10% of the maximum that the lake can support?

Biology Environment 3 Variables Exponential High

Source: adapted from Bolker USING ALG pages 191-194

197. The concentration of drugs in the bloodstream decreases exponentially over time. For Gigamycin the half-life is 5.2 hours; half of the original dosage level is still present after 5.2 hours. At noon a patient receives an injection that brings his blood Gigamycin concentration to 6.5 micrograms per liter. A second injection is to be administered when the concentration falls to 1.4 micrograms per liter. When will that be?

Biology Medicine Two Variables Exponential High

Source: Bolker USING ALG page 163

198. The Trauma Score is a measure of injury severity, and is based on seven assessments that medical professionals can obtain easily. A 'probability of survival' can then be estimated from the Trauma Score by the following function:

where P_s(T) is the probability (from 0.000 to 1.000) of survival, e is the base of the natural logarithms (about 2.718), and T is the trauma score (whole numbers ranging from 1 to 16). Use this function to estimate the probability of survival for a new patient with a Trauma Score of 14. Also, use the function to estimate the Trauma Score for patients with a probability of survival of 0.90.

Biology Medicine Two Variables Exponential High

Source: adapted from Sacco MATHEMATICS AND MEDICINE pages 4-17

199. A population of a certain virus grows so that its size S after t days is given by: , where S₀ is the original size.

If the original size is 1000, what will the size be at the end of 15 and 30 days? How long does it take the population to double?

Biology Medicine Two Variables Exponential High

Source: adapted from Farlow APPLIED MATH page 662

200. On a college campus of 5000 students, one student returned from vacation with a very contagious flu virus. The spread of the virus through the student body is given by , where s is the number infected after t days; the number s is the cumulative number infected.

How many are infected after five days? After how many days will 40% or more of the students have been infected?

Biology Medicine Two Variables Exponential High

Source: adapted from Larson ALG & TRIG page 361, with errors fixed

201. Impulses in nerve fibers travel at a speed of 293 ft/sec. Therefore, the distance D traveled in t seconds is given by D = 293t. How long would it take an impulse to travel from the brain to the toes of a person who is 6 feet tall? What is the fastest reaction time to apply brakes possible for a person who is 5 ft 8 in tall, if it is about 4 ft 10 in from the brain to the furthest muscle needed to apply the brakes?

Biology Medicine and Health Two Variables First Degree Medium

Source: based on Bittinger APPLIED CALCULUS page 47

202. The equation expresses the cost y (in thousands of dollars) of removing x% of a certain pollutant from the atmosphere of a large city. The graph of such a function is called a cost-benefit curve. (Note that removing 10% means x=10, and that the constants 20 and 104 depend on the nature of the pollutant and the city ecology, and these numbers change over time.) Find the cost of removing 40%, 80%, 90%, 95% and 100% of the pollutant. If a federal grant allows the city to budget $300,000 to remove the pollutant, what percent of the pollutant can be removed?

Biology Environment Two Variables Rational High

Source: adapted from Piascik COLLEGE MATH page 112

203. On September 1, researchers from the Center for Disease Control estimated that the number of active cases of German Measles in New York City was 200. In addition to a rate of new cases equal to 20% of the current infected population per month, there are also 100 active cases per month from immigration. (The rate of new cases takes into account a higher rate of infection, but is lower because of those who recover from the infection.) Based on these facts, the researchers can predict the number of active cases by using the formula:

where t is measured in months.

If an epidemic will be declared when the number of active cases is 600, when can they expect to see an epidemic?

Biology Medicine Two Variables Exponential High Source: LCC

204. A cougar spots a fawn 132 meters away. The cougar starts toward the fawn at a speed of 18 meters per second (m/sec). At the same instant, the fawn starts running away at 11 m/sec. The cougar has enough energy to run for 17 seconds. Will it catch the fawn?

Biology Wildlife One Variable First Degree High

Source: Foerster Algebra I pg 139

205. The safe dosage of a new medicine is determined by testing it on animals. To predict the safe dosage for humans from the results of animal experiments, scientists assume that the safe dosage is directly proportional to the patient's skin area. This area is directly proportional to the square of the person's height (assuming that patients have the same proportions). These statements mean that   d = kh², where d is the safe dosage, k is a constant, and h is a person's height. Suppose that a new cold remedy is tested on monkeys 30 cm tall, and that the safe dosage for the monkeys is found to be 1.2 milligrams. Find the equation relating the dosage to the height. Assuming that humans have roughly the same proportions as monkeys, predict the safe dosage of this cold remedy for a 90 cm tall child, a 170 cm tall adult (average), and a 2 m tall adult.

Biology Medicine Two Variables Second Degree High

Source: from Foerster ALGEBRA & TRIG page 220

206. The radiation from a surface depends on the fourth power of the temperature. The formula: , where is the Stefan-Boltzmann constant, e is the emissivity of the object, and T is the temperature in degrees Kelvin. Water freezes at 273K, and .

In the operation of the thermograph, the radiation of each small area of the skin is measured. Because the skin over a tumor is warmer than elsewhere, thermograms are widely used in screening for breast cancer. What is the percentage difference between the radiation rates from skin at 34 and 35C?

Biology Medicine Two Variables 3rd Degree High

Source: adapted from Beiser APPLIED PHYSICS page 189-191

207. The home range of an animal is defined as the region to which the animal confines its movements. It has been hypothesized in statistical studies (J. M. Emlen, ECOLOGY: AN EVOLUTIONARY APPROACH, page 200, Addison-Wesley, 1973) that the area H of that region can be approximated using the body weight W of an animal by the function . The territorial area is defined to be its defended region, or exclusive region. It has been hypothesized that the area T of that region can be approximated by the function . Find the home range and territorial area for animals with weights of 20, 40, 60 and 80.

Biology Wildlife Two Variables Exponential Medium

Source: Bittinger APPLIED CALCULUS pages 59 and 65

208. The expected weight W (in tons) of an adult humpback whale is related to its length L (in feet) by the linear equation W = 1.70L - 42.8. Estimate the weight of a 30-foot humpback whale. If the error in estimating the length could be as large as 2 feet, what is the corresponding error for the weight estimate?

Biology Wildlife One Variable First Degree Medium

Source: Swokowski, page 141

209. To diagnose a certain disease, a tracer dye is injected into the pancreas. A healthy person will secrete 5% of the dye remaining in the pancreas each minute. For one test, 0.3 grams is injected, and the amount D present after t minutes for a healthy person is predicted by:

This particular person had 0.1 gram present after 30 minutes. How does this compare to what a healthy person would have had?

Biology Medicine Two Variables Exponential Medium

Source: from Varberg APPLIED CALC page 195

210. The intensity of sound varies inversely with the square of the distance. A certain person speaking normally produces a sound intensity of 40 dB (decibels) at a distance of 3 feet. If the threshold intensity of reasonable audibility is 20 dB, how far away can the person be heard clearly? (Note that a increase of 10 dB is increase by a factor of 10 in the intensity.)

Biology Medicine Two Variables Logarithmic Medium

Source: adapted from Beiser APPLIED PHYSICS page 137

211. The Count Model is am empirically based formula that can be used to predict the height of a preschooler. If h denotes the height (cm) at age x (years) between ¼ and 6, then h can be approximated by . Predict the height when a child reaches age 2, and predict the age for the child whose height is 100 cm.

Biology Medicine and Health Two Variables Logarithmic Medium

Source: from Swokowski CALCULUS page 389

212. The Ehrenberg Relation, is an empirically based formula relating the height h (cm) to the weight (kg) for children aged 5 through 13. The formula, with minor changes in the constants, has been verified in many different countries. Estimate the weight of a child who is 150 cm tall, and estimate the height of a child who weighs 50 kg.

Biology Medicine and Health Two Variables Logarithmic Medium

Source: from Swokowski CALCULUS page 391

213. In physiology, experiments suggest that the relationship of loudness and intensity of sound is a logarithmic one known as the Weber-Fechner law: The apparent loudness L of a sound is proportional to the natural logarithm of its actual intensity. What actual increase in intensity will cause a doubling of the apparent loudness? If the intensity is doubled, what is the apparent change in loudness?

Biology Medicine Two Variables Logarithmic Medium

Source: Gustafson COLLEGE ALG pages 371 and 373

214. The amount of energy that a salmon expends in swimming upstream with velocity v (relative to the stream) over a period of time T has been shown experimentally to be: E = cv³T, where c is a constant. If the velocity of the stream is 4 miles per hour and the salmon swims upstream 200 miles, the equation becomes:

(v>4)

Choose some values for the constant c, such as c=2 or c=3, and graph the resulting function. From the graphs, what is the velocity that will minimize the energy expended? Does this value of the velocity change with different values of c?

(Note to instructors: Reality matches the model; most salmon do swim at a velocity roughly 50% greater than the velocity of the current.

Biology Wildlife Two Variables Rational Medium

Source: Adapted from Farlow APPLIED MATH page 614-615

215. The number of students in an elementary school who will get the flu t days after the first student shows symptoms is given by the formula N(t) = 40t - 5t². If a fourth grader comes down with the flu on Monday, on what day will the largest number of students have the flu? How many will have the flu on that day?

Biology Medicine Two Variables Second Degree Medium Source: McKeague Interm pg 576

216. Each year, many states allow hunters to shoot deer during a limited open season whose length is carefully chosen to ensure a harvest that is sustainable year after year. For one state, the yearly growth curve for the deer population is estimated to be y = 1.4x - 0.0004x², where x is the population in thousands. Write an equation to represent the population after a harvest of x thousand deer. Graph both curves to find the largest population and the annual harvest that it will sustain.

Biology Wildlife Two Variables Second Degree Medium

Source: from Varberg APPLIED CALC page 141

217. Blood pressure in a patient will drop by an amount D, where D = 0.025x²(30 - x) and x is the amount of a drug injected. The dropping rate, RD, of the blood pressure is given by RD = 1.5x - 0.075x². Graph both functions, and find the dosage that provides the greatest drop in blood pressure.

Biology Medicine 3 Variables 3rd Degree Medium

Source: adapted from Farlow APPLIED MATH page 621

218. Carbon monoxide contains 43% carbon, while carbon dioxide is only 27% carbon. The Environmental Protection Agency (EPA) finds that a 1600-mg sample of exhaust gas has 32% carbon. Assuming that all carbon comes from carbon monoxide or carbon dioxide, how much of each is in the sample?

Biology Environment Two Variables First Degree Low

Source: adapted from Foerster Algebra I pg 326

219. Stream C, running at 12 cubic feet per second, is fed by two other streams, A and B. Stream A contributes 4 cubic feet per second, and stream B adds 8 cubic feet per second. The dissolved oxygen is 4% in stream A and 6% in stream B. The fish in stream C need 5.5% dissolved oxygen in order to survive. How much dissolved oxygen is available in stream C? Will the fish survive?

Biology Wildlife One Variable First Degree Low

Source: Saunders WHEN ARE WE EVER GONNA page 120

The problems listed were taken from some of the references listed below. Other problems are in these references, and some references listed below were not cited in the listing above.

Aufmann, Richard & Barker, Vernon INTRODUCTORY ALGEBRA 3rd edition Houghton Mifflin, 1991

Agnew, Jeanne in the AIM (Applications in Mathematics) series:

all published by the Mathematical Association of America, 1986

Barros-Neto, José COLLEGE ALGEBRA WITH APPLICATIONS   West Publishing, 1985

Beiser, Arthur THEORY AND PROBLEMS OF APPLIED PHYSICS 2nd edition   McGraw-Hill (Schaum's Outline Series), 1988

Berkey, Dennis APPLIED CALCULUS         Saunders College Publishing, 1987

Bittinger, Marvin and Morrel, Bernard APPLIED CALCULUS 2nd edition    Addison-Wesley, 1988

Bolker, Ethan USING ALGEBRA          Wyndham Hall Press, 1990

Center for Occupational Research and Development (CORD) APPLIED MATHEMATICS (series of modules designed for Tech-Prep programs)         CORD, Waco TX, 1988

Demana, Franklin & Leitzel, Joan TRANSITION TO COLLEGE MATHEMATICS         Addison Wesley, 1984

Demana, Franklin & Waits, Bert & Clemens, Stanley COLLEGE ALGEBRA & TRIGONOMETRY A GRAPHING APPROACH, 2nd edition       Addison-Wesley, 1992

Faber, Richard & Freedman, Marvin & Kaplan, James APPLIED CALCULUS, AN INTUITIVE APPROACH FOR MANAGEMENT, LIFE, AND SOCIAL SCIENCES      West Publishing, 1986

Farlow, Stanley & Haggard, Gary APPLIED MATHEMATICS FOR MANAGEMENT, LIFE SCIENCES, AND SOCIAL SCIENCES        Random House, 1988

Foerster, Paul A   ALGEBRA I, EXPRESSIONS, EQUATIONS AND APPLICATIONS   and       ALGEBRA AND TRIGONOMETRY, FUNCTIONS AND APPLICATIONS        Addison Wesley, 1984

Globe MATHEMATICS WORKSHOP: EXPLORING CAREERS         Globe Book Company, 1990

Gustafson, Daniel PHYSICS: HEALTH AND THE HUMAN BODY       Wadsworth Publishing, 1980

Gustafson, David and Frisk, Peter COLLEGE ALGEBRA 4th edition       Brooks/Cole Publishing, 1990

Hestenes, Marshall and Hill, Richard ALGEBRA AND TRIGONOMETRY 2nd edition       Prentice-Hall, 1986

Larson, Roland & Hostetler, Robert ALGEBRA AND TRIGONOMETRY 2nd edition       D. C. Heath, 1989

LCC Lansing Community College Faculty    (Howard Jones and James Stewart in particular)

McKeague, Charles P INTERMEDIATE ALGEBRA, A TEXT/WORKBOOK 3rd edition  Harcourt Brace Jovanovich, 1990

MTH 102 Course Packet APPLICATIONS SUPPLEMENTS   Lansing Community College Math Lab, 1990

NCTM APPLICATIONS OF SECONDARY SCHOOL MATHEMATICS      edited by Joe Dan Austin, NCTM 1991

Piascik, Chester COLLEGE MATHEMATICS with Applications to Management, Economics, and the Social and Natural Sciences          Charles E. Merrill Publishing, 1984

Pulsinelli, Linda & Hooper, Patricia INTRODUCTORY ALGEBRA, AN INTERACTIVE APPROACH     MacMillan, 1991

Ruud, Warren and Shell, Terry PRELUDE TO CALCULUS         Wadsworth Publishing, 1990

Rotman, Jack          Created/generated by the author

Sacco, William; Copes, Wayne; Sloyer, Clifford; and Stark, Robert MATHEMATICS AND MEDICINE: HOW SERIOUS IS THE INJURY?          Janson Publications, 1987        (part of "Contemporary Applied Mathematics" Series)

Saunders, Hal WHEN ARE WE EVERY GONNA HAVE TO USE THIS?, 3rd edition       Dale Seymour Publications, 1988

Sloyer, Cliff FANTASTIKS OF MATHEMATICS, Applications of Secondary Mathematics      Janson Publications, 1986

Swokowski, Earl FUNDAMENTALS OF ALGEBRA AND TRIGONOMETRY, 7th edition       PWS Kent, 1989

Swokowski, Earl CALCULUS, 6th edition   PWS Kent, 1991

UCSMP (University of Chicago School Mathematics Project)    ALGEBRA , ADVANCED ALGEBRA    both published by Scott, Foresman & Company, 1990

Varberg, Dale and Fleming, Walter APPLIED CALCULUS FOR MANAGEMENT, SOCIAL, AND LIFE SCIENCES  Prentice-Hall, 1991

Washington, Allyn & Triola, Mario INTRODUCTION TO TECHNICAL MATHEMATICS, 3rd edition       Benjamin/Cummings, 1984

Washington, Allyn BASIC TECHNICAL MATHEMATICS WITH CALCULUS, 4th edition    Benjamin/Cummings, 1985

1. © Copyright 1992, Lansing Community College.