2.3

Problem Solving and Applications of Linear Equations 2.3

OBJECTIVES 1. Apply the five-step strategy for solving an application 2. Solve motion problems 3. Find a break-even point 4. Solve a literal equation

“I have a problem!” How often have you, or a friend, started a conversation with this statement? And, more important, what do you do to solve the problem? George Polya, in his book How to Solve It, contends that there are four parts to solving any problem. 1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

This approach is useful for solving any problem. One of the reasons that mathematics is a required course in most programs is that, in a math class, you are constantly practicing the art of problem solving. To help you remember to use Polya’s approach on problems that you encounter outside of the classroom, we will consistently use a series of steps similar to those prescribed above. Recall that an algorithm is a series of steps that, when followed, solves a problem. The following five-step algorithm is essentially the same as Polya’s approach, but for one thing. We have divided the “devise a plan” step into two steps that are more directly relevant to solving mathematical problems. Here is our five-step approach:

Step by Step: Solving Applications

© 2001 McGraw-Hill Companies

Step 1 Read the problem carefully to determine the unknown quantities. Step 2 Choose a variable to represent the unknown. Express all other unknowns in terms of this variable. Step 3 Translate the problem to the language of algebra to form an equation. Step 4 Solve the equation, and answer the question of the original problem. Step 5 Verify your solution by returning to the original problem.

Our first two applications fall into the category of uniform-motion problems. Uniform motion means that the speed of an object does not change over a certain distance or time. To solve these problems, we will need a relationship between the distance traveled, represented by d, the rate (or speed) of travel, r, and the time of that travel, t. In general, the relationship for the distance traveled d, rate r, and time t, is expressed as d Distance



r Rate



t Time

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LINEAR EQUATIONS AND INEQUALITIES

This is the key relationship, and it will be used in all motion problems. Let’s see how it is applied in Example 1.

Example 1 Solving a Motion Problem On Friday morning Ricardo drove from his house to the beach in 4 h. In coming back on Sunday afternoon, heavy traffic slowed his speed by 10 mi/h, and the trip took 5 h. What was his average speed (rate) in each direction? Step 1 We want the speed or rate in each direction. Step 2 Let x be Ricardo’s speed to the beach. Then x  10 is his return speed. It is always a good idea to sketch the given information in a motion problem. Here we would have x mi h for 4 h

Going

mi

x  10 h for 5 h

Returning

Step 3 Because we know that the distance is the same each way, we can write an equation, using the fact that the product of the rate and the time each way must be the same. So Distance (going)  distance (returning) Time  rate (going)  time  rate (returning) 4x  5(x  10)



Time  rate (going)

Time  rate (returning)

A chart can help summarize the given information. We begin by filling in the information given in the problem.

Rate

Time

Going

x

4

Returning

x  10

5

Distance

Now we fill in the missing information. Here we use the fact that d  rt to complete the chart.

Rate

Time

Distance

Going

x

4

4x

Returning

x  10

5

5(x  10)

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CHAPTER 2



70

PROBLEM SOLVING AND APPLICATIONS OF LINEAR EQUATIONS

SECTION 2.3

71

From here we set the two distances equal to each other and solve as before. Step 4 Solve. 4x  5(x  10) 4x  5x  50 x  50 x  50 mi/h NOTE x was his rate going; x  10, his rate returning.

So Ricardo’s rate going to the beach was 50 mi/h, and his rate returning was 40 mi/h. Step 5 To check, you should verify that the product of the time and the rate is the same in each direction.

CHECK YOURSELF 1 A plane made a flight (with the wind) between two towns in 2 h. Returning against the wind, the plane’s speed was 60 mi/h slower, and the flight took 3 h. What was the plane’s speed in each direction?

Example 2 illustrates another way of using the distance relationship.

Example 2 Solving a Motion Problem Katy leaves Las Vegas for Los Angeles at 10 A.M., driving at 50 mi/h. At 11 A.M. Jensen leaves Los Angeles for Las Vegas, driving at 55 mi/h along the same route. If the cities are 260 mi apart, at what time will they meet? Step 1 Let’s find the time that Katy travels until they meet. Step 2 Let x be Katy’s time. Then x  1 is Jensen’s time. Jensen left 1 h later! Again, you should draw a sketch of the given information.

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(Jensen) 55 mi h for (x  1) h

(Katy) 50 mi h for x h

Los Angeles

Las Vegas Meeting point

Step 3 To write an equation, we will again need the relationship d  rt. From this equation, we can write Katy’s distance  50x Jensen’s distance  55(x  1)

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CHAPTER 2

LINEAR EQUATIONS AND INEQUALITIES

As before, we can use a table to solve.

Rate

Time

Distance

Katy

50

x

50x

Jensen

55

x1

55(x  1)

From the original problem, the sum of those distances is 260 mi, so 50x  55(x  1)  260 Step 4 50x  55(x  1)  260 50x  55x  55  260 105x  55  260 105x  315 x3h question asked in the problem.

Finally, because Katy left at 10 A.M., the two will meet at 1 P.M. We leave the check of this result to you. CHECK YOURSELF 2 At noon a jogger leaves one point, running at 8 mi/h. One hour later a bicyclist leaves the same point, traveling at 20 mi/h in the opposite direction. At what time will they be 36 mi apart?

Next, we consider an application from business. But we will need some new terminology. The total cost of manufacturing an item consists of two types of costs. The fixed cost, sometimes called the overhead, includes costs such as product design, rent, and utilities. In general, this cost is constant and does not change with the number of items produced. The variable cost, which is a cost per item, includes costs such as material, labor, and shipping. The variable cost depends on the number of items being produced. A typical cost equation might be C



3.30x

Variable cost



5000 Fixed cost

in which total cost, C, equals variable cost times the number of items produced, x, plus the fixed cost. The total revenue is the income the company makes. It is calculated as the product of the selling price of the item and the number of items sold. A typical revenue equation might be R



7.50x

Selling price per item

Number of items sold

and total revenue equals an item’s selling price times the number sold.

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NOTE Be sure to answer the

PROBLEM SOLVING AND APPLICATIONS OF LINEAR EQUATIONS

SECTION 2.3

73

The break-even point is that point at which the revenue equals the cost (the company would exactly break even without a profit or a loss). Let’s apply these concepts in Example 3. Example 3 Finding the Break-Even Point A firm producing DVDs finds that its fixed cost is $5000 per month and that its variable cost is $3.50 per DVD. The cost of producing x DVDs is then given by

C  3.50x  5000 The firm can sell the units at $7.50 each, so the revenue from selling x units is R  7.50x Find the break-even point. Because the break-even point is that point at which the revenue equals the cost, or R  C, from our given equations we have 

3.50x  5000



7.50x Revenue

Cost

Solving as before gives 4x  5000 x  1250 The firm will break even (no profit or loss) by producing and selling exactly 1250 DVDs each month.

© 2001 McGraw-Hill Companies

CHECK YOURSELF 3 A firm producing lawn chairs has fixed costs of $525 per week. The variable cost is $8.50 per chair, and the revenue per chair is $15.50. This means that the cost equation is

C  8.50x  525 and the revenue equation is

R  15.50x Find the break-even point.

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Let’s turn now to an application of our work in solving literal equations for a specified variable.

Example 4 Solving an Interest Rate Application

NOTE From the given problem,

Suppose a principal of $1000 is invested in a mutual fund account for 5 years. For the amount in the account to be $1400 at the end of that period, what must the interest rate be? In Example 3 of Section 2.2, we solved the formula

we know values for A, P, and t. Two strategies are possible.

A  P(1  rt) for r with the result r

AP Pt

In our problem, A  $1400, P  $1000, and t  5 years. Substituting, we have r

1400  1000 400   0.08 (1000)(5) 5000

The necessary interest rate is 8%.

CHECK YOURSELF 4 Suppose that a principal of $5000 is invested in a time deposit fund for 3 years. If the amount in the account at the end of that period is $6050, what was the annual interest rate?

Note, in the previous example and exercise, that once we solved for r in the original equation, we were able to easily use the result with different sets of data. You can compare this to first substituting the known values and then having to solve for r in each case separately. Solving for a specified variable also has significance in computer programming. In using a formula for the computation of the value of a variable, you must solve the formula for that variable to use the formula in a program.

CHECK YOURSELF ANSWERS 1. 180 4. 7%

mi mi with the wind and 120 against h h

2. At 2 P.M.

3. 75 chairs © 2001 McGraw-Hill Companies

1. We can substitute the known values in A  P(1  rt) and then solve for r. 2. Or we can solve for r and then substitute. We have illustrated this approach.

Name

2.3

Exercises

Section

Date

Solve the following applications. 1. Speed. On her way to a business meeting, Kim took the freeway, and the trip took

3 h. Returning, she decided to take a side road, and her speed along that route 1 averaged 9 mi/h slower than on the freeway. If Kim’s return trip took 3 h and the 2 distance driven was the same each way, find her average speed in each direction.

ANSWERS

1. 2.

2. Speed. Beth was required to make a cross-country flight in training for her pilot’s

license. When she flew from her home airport, a steady 30-mi/h wind was behind her, and the first leg of the trip took 5 h. When she returned against the same wind, the flight took 7 h. Find the plane’s speed in still air and the distance traveled on each leg of the flight.

3. Speed. Craig was driving on a 220-mi trip. For the first 3 h he traveled at a steady

speed. At that point, realizing that he would be late to his destination, he increased his speed by 10 mi/h for the remaining 2 h of the trip. What was his driving speed for each portion of the trip?

3. 4. 5. 6. 7. 8.

4. Distance. Robert can drive to work in 45 min, whereas if he decides to take the bus,

the same trip takes 1 h 15 min. If the average rate of the bus is 16 mi/h slower than his driving rate, how far does he travel to work?

5. Time. At 9 A.M., Tom left Boston for Baltimore, traveling at 45 mi/h. One hour later,

Andrea left Baltimore for Boston, traveling at 50 mi/h along the same route. If the cities are 425 mi apart, at what time did Tom and Andrea meet?

© 2001 McGraw-Hill Companies

6. Time. A passenger bus left a station at 1 P.M., traveling north at an average rate of

50 mi/h. One hour later, a second bus left the same station, traveling south at a rate of 55 mi/h. At what time will the two buses be 260 mi apart? 7. Distance. On Tuesday, Malia drove to a conference and averaged 54 mi/h for the

trip. When she returned on Thursday, road construction slowed her average speed by 9 mi/h. If her total driving time was 11 h, what was her driving time each way, and how far away from her home was the conference? 8. Time. At 8:00 A.M., Robert left on a trip, traveling at 45 mi/h. One-half hour later,

Laura discovered that Robert forgot his luggage and left along the same route, traveling at 54 mi/h, to catch up with him. When did Laura catch up with Robert? 75

ANSWERS 9.

9. Business. A firm producing gloves finds that its fixed cost is $4000 per week and its

variable cost is $8.50 per pair. The revenue is $13.50 per pair of gloves, so that cost and revenue equations are, respectively,

10. 11. 12. 13.

C  8.50x  4000

and

R  13.50x

Find the break-even point for the firm.

10. Business. A company that produces calculators determines that its fixed cost is

$8820 per month. The variable cost is $70 per calculator: the revenue is $105 per calculator. The cost and revenue equations, respectively, are given by C  70x  8820

and

R  105x

Find the number of calculators the company must produce and sell to break even.

11. Business. A firm that produces scientific calculators has a fixed cost of $1260 per

week and variable cost of $6.50 per calculator. If the company can sell the calculators for $13.50, find the break-even point.

12. Business. A publisher finds that the fixed cost associated with a new paperback is

$18,000. Each book costs $2 to produce and will sell for $6.50. Find the publisher’s break-even point.

13. Business. A firm producing flashlights finds that its fixed cost is $2400 per week,

C  4.50x  2400

and

R  7.50x

Find the break-even point for the firm (the point at which the revenue equals the cost). 76

© 2001 McGraw-Hill Companies

and its variable cost is $4.50 per flashlight. The revenue is $7.50 per flashlight, so the cost and revenue equations are, respectively,

ANSWERS

14. Business. A company that produces portable television sets determines that its fixed

14.

cost is $8750 per month. The variable cost is $70 per set, and the revenue is $105 per set. The cost and revenue equations, respectively, are given by

15.

C  70x  8750

16.

and

R  105x

Find the number of sets the company must produce and sell to break even. 17. 15. Business. An important economic application involves supply and demand. The

number of units of a commodity that manufacturers are willing to supply, S, is related to the market price, p. A typical supply equation is S  40p  285

(1)

(Generally the supply increases as the price increases.) The number of units that consumers are willing to buy, D, is called the demand, and it is also related to the market price. A typical demand equation is D  45p  1500

(2)

(Generally the demand decreases as the price increases.) The price at which the supply and demand are equal (or S  D) is called the equilibrium price for the commodity. The supply and demand equations for a certain model portable radio are given in equations (1) and (2). Find the equilibrium price for the radio. 16. Business. The supply and demand equations for a certain type of computer modem

are

© 2001 McGraw-Hill Companies

S  25p  2500

and

D  40p  5300

Find the equilibrium price for the modem. 17. Business. You find a new bicycle that you like, and you plan to ride it for exercise

several days a week. You are also happy to find that this very model is on sale for 22% off. You speak to the salesclerk, who begins writing up the sale by first adding on your state’s 7.8% sales tax. He then takes off the 22%. “No!”, you object, “You should take the 22% off first and then add the sales tax.” The salesclerk says he is sorry, but he has been instructed to first calculate the amount of tax. Who is correct? Defend your position using algebra.

77

ANSWERS

18. Density and Body Fat. You have probably heard the question, “Which weighs more,

a ton of feathers or a ton of bricks?” The usual response of “bricks” is, of course, incorrect. A ton of anything weighs 2000 pounds. Why do people miss this question so often? We know from experience that bricks are heavier than feathers. However, what we mean is that bricks have a much higher density than feathers (they weigh more per unit volume). A cubic meter of bricks weighs far more than a cubic meter of feathers. We often compare the density of a substance to the density of water. Water has a density of 1 g/cm3. Things denser than water (such as body muscle and bone) sink, whereas things less dense (such as body fat) float. The concept of density is important in analyzing the significance of the percentage of body fat you have. There are many ways to measure, or estimate, your percentage of body fat. Most health clubs and many college health classes offer the opportunity to find this percentage. By multiplying percentage of body fat times your weight, you can calculate the actual pounds of fat in your body. However, we are looking for a more useful measure. Body density, D, can be calculated using the following formula.

18.

D

1 A B  a b

in which A  proportion of body fat B  proportion of lean body tissue (1  A) a  density of fat body tissue in grams/cubic cm (approximately 0.9 g/cm3) b  density of lean body tissue in grams/cubic cm (approximately 1.1 g/cm3) Use this formula to compute the following. 1. Substituting 1  A for B, 0.9 for a, and 1.1 for b, solve the formula so that D is expressed in terms of A. 2. Find the values associated with body fat proportions of 0.1, 0.15, 0.2, 0.25, and 0.3. 3. Use other resources (health professionals, health clubs, the Internet, and the library) to find another measure of body density. How do the two methods compare?

Answers

© 2001 McGraw-Hill Companies

1. 63 mi/h going; 54 mi/h returning 3. 40 mi/h for 3 h; 50 mi/h for 2 h 5. 2:00 P.M. 7. 5 h going; 6 h returning; 270 mi 9. 800 pairs of gloves 11. 180 calculators 13. 800 flashlights 15. $21 17.

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