19 (3-56) Chapter 3 Linear Equations in Two Variables and Their Graphs Height (feet) a) Find h(35) to the nearest tenth of an inch. 19.1 feet b) Who gains more height from an increase of 1 ft/sec in velocity: a fast runner or a slow runner? Fast runner 25 2 15 1 5 1 2 3 4 Velocity (ft/sec) FIGURE FOR EXERCISE 84 85. Credit card fees. A certain credit card company gets 4% of each charge, and the retailer receives the rest. At the end of a billing period the retailer receives a statement showing only the retailer s portion of each transaction. Express the original amount charged C as a function of the retailer s portion r. r C. 96 86. More credit card fees. Suppose that the amount charged on the credit card in the previous exercise includes 8% sales tax. The credit card company does not get any of the sales tax. In this case the retailer s portion of each transaction includes sales tax on the original cost of the goods. Express the original amount charged C as a function of the retailer s portion. C 2 7 R 26 GETTING MORE INVOLVED Discussion. In each situation determine whether a is a function of b, b is a function of a, or neither. Answers may vary depending on interpretations. 87. a the price per gallon of regular unleaded. b the number of gallons that you get for $1. Both 88. a the universal product code of an item at Sears. b the price of that item. b is a function of a 89. a a student s score on the last test in this class. b the number of hours he/she spent studying. Neither 9. a a student s score on the last test in this class. b the IQ of the student s mother. Neither 91. a the weight of a pacage shipped by UPS. b the cost of shipping that pacage. Neither 92. a the Celsius temperature at any time. b the Fahrenheit temperature at the same time. Both 93. a the weight of a letter. b the cost of mailing the letter. b is a function of a 94. a the cost of a gallon of mil. b the amount of sales tax on that gallon. b is a function of a 3.7 VARIATION In this section If y 5x, the value of y depends on the value of x. As x varies, so does y. Certain functions are customarily expressed in terms of variation. In this section you will learn to write formulas for those functions from verbal descriptions of the functions. Direct Variation Finding the Constant Inverse Variation Joint Variation Direct Variation Suppose you average 6 miles per hour on the freeway. The distance D that you travel depends on the amount of time T that you travel. Using the formula D R T, we can write D 6T. Consider the possible values for T and D given in Table 3.3. T (hours) 1 2 3 4 5 6 D (miles) 6 12 18 24 3 36 TABLE 3.3
3.7 Variation (3-57) 191 D Distance (miles) 4 3 2 1 1 2 3 4 5 6 Time (hours) T FIGURE 3.33 The graph of D 6T is shown in Fig. 3.33. Note that as T gets larger, so does D. In this situation we say that D varies directly with T, or D is directly proportional to T. The constant rate of 6 miles per hour is called the variation constant or proportionality constant. Notice that D is simply a linear function of T. We are just introducing some new terms to express an old idea. s Direct Variation The statement y varies directly as x or y is directly proportional to x means that y x for some constant. The constant of variation is a fixed nonzero real number. Finding the Constant If we now one ordered pair in a direct variation, then we can find the constant of variation. E X A M P L E 1 helpful hint In any variation problem you must first determine the general form of the relationship. Because this problem involves direct variation, the general form is y x. Finding a constant of variation Natasha is traveling by car, and the distance D that she travels varies directly as the rate R at which she drives. At 45 miles per hour, Natasha travels 135 miles. Find the constant of variation, and write D as a function of R. Because D varies directly as R, there is a constant such that D R. Because D 135 when R 45, we can write 135 45 or 3. So D 3R.
192 (3-58) Chapter 3 Linear Equations in Two Variables and Their Graphs In the next example we find the constant of variation and use it to solve a variation problem. E X A M P L E 2 A direct variation problem Your electric bill at Middle States Electric Co-op varies directly with the amount of electricity that you use. If the bill for 28 ilowatts of electricity is $196, then what is the bill for 4 ilowatts of electricity? Because the amount A of the electric bill varies directly as the amount E of electricity used, we have A E for some constant. Because 28 ilowatts cost $196, we have 196 28 or.7. So A.7E. Now if E 4 we get A.7(4) 28. The bill for 4 ilowatts would be $28. Inverse Variation If you plan to mae a 4-mile trip by car, the time it will tae depends on your rate of speed. Using the formula D RT, we can write 4 T. R Consider the possible values for R and T given in the following table: R (mph) 1 2 4 5 8 1 T (hours) 4 2 1 8 5 4 The graph of T 4 is shown in Fig. 3.34. As your rate increases, the time for the R trip decreases. In this situation we say that the time is inversely proportional to the speed. T 4 Time (hours) 3 2 1 2 4 6 8 1 R Rate (miles per hour) FIGURE 3.34
3.7 Variation (3-59) 193 Inverse Variation The statement y varies inversely as x, or y is inversely proportional to x means that y x for some nonzero constant of variation. CAUTION The constant of variation is usually positive because most physical examples involve positive quantities. However, the definitions of direct and inverse variation do not rule out a negative constant. E X A M P L E 3 P 2 g/cm 2 P 15 g/cm 2 V 12 cm 3 V? FIGURE 3.35 An inverse variation problem The volume of a gas in a cylinder is inversely proportional to the pressure on the gas. If the volume is 12 cubic centimeters when the pressure on the gas is 2 ilograms per square centimeter, then what is the volume when the pressure is 15 ilograms per square centimeter? See Fig. 3.35. Because the volume V is inversely proportional to the pressure P, we have V P for some constant. Because V 12 when P 2, we can find by substituting these values into the above formula: 12 2 2 12 2 Multiply each side by 2. 2 24 Now to find V when P 15, we can use the formula V 24 : P 24 V 15 16 So the volume is 16 cubic centimeters when the pressure is 15 ilograms per square centimeter. Joint Variation If the price of carpet is $3 per square yard, then the cost C of carpeting a rectangular room depends on the width W (in yards) and the length L (in yards). As the width or length of the room increases, so does the cost. We can write the cost as a function of the two variables L and W: C 3LW We say that C varies jointly as L and W.
194 (3-6) Chapter 3 Linear Equations in Two Variables and Their Graphs Joint Variation The statement y varies jointly as x and z or y is jointly proportional to x and z means that y xz for some nonzero constant of variation. E X A M P L E 4 helpful hint Because the variation in this problem is joint, we now the general form of the function is y xz, where is the constant of variation. A joint variation problem The cost of shipping a piece of machinery by truc varies jointly with the weight of the machinery and the distance that it is shipped. It costs $3 to ship a 25-lb milling machine a distance of 6 miles. Find the cost for shipping a 15-lb lathe a distance of 8 miles. Because the cost C varies jointly with the weight w and the distance d, we have C wd where is the constant of variation. To find, we use C 3, w 25, and d 6: 3 25 6 3 Divide each side by 25 6. 25 6.2 Now use w 15 and d 8 in the formula C.2wd: C.2 15 8 24 So the cost of shipping the lathe is $24. CAUTION The variation words (directly, inversely, or jointly) are never used to indicate addition or subtraction. We use multiplication in the formula unless we see the word inversely. We use division only for inverse variation. WARM-UPS True or false? Explain your answer. 1. If y varies directly as z, then y z for some constant. True 2. If a varies inversely as b, then a b for some constant. False 3. If y varies directly as x and y 8 when x 2, then the variation constant is 4. True 4. If y varies inversely as x and y 8 when x 2, then the variation constant is 1 4. False 5. If C varies jointly as h and t, then C ht. False 6. The amount of sales tax on a new car varies directly with the purchase price of the car. True 7. If z varies inversely as w and z 1 when w 2, then z 2. True w
3.7 Variation (3-61) 195 WARM-UPS (continued) 8. The time that it taes to travel a fixed distance varies inversely with the rate. True 9. If m varies directly as w, then m w for some constant. False 1. If y varies jointly as x and z, then y (x z ) for some constant. False 3. 7 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What does it mean to say that y varies directly as x? If y varies directly as x, then there is a constant such that y x. 2. What is a variation constant? A variation constant is the constant in the formulas y x or y x. 3. What does it mean to say that y is inversely proportional to x? If y is inversely proportional to x, then there is a constant such that y x. 4. What does it mean to say that y varies jointly as x and z? If y varies jointly as x and z, then there is a constant such that y xz. Write a formula that expresses the relationship described by each statement. Use for the constant in each case. See Examples 1 4. 5. T varies directly as h. T h 6. m varies directly as p. m p 7. y varies inversely as r. y r 8. u varies inversely as n. u n 9. R is jointly proportional to t and s. R ts 1. W varies jointly as u and v. W uv 11. i is directly proportional to b. i b 12. p is directly proportional to x. p x 13. A is jointly proportional to y and m. A ym 14. t is inversely proportional to e. t e Find the variation constant, and write a formula that expresses the indicated variation. See Example 1. 15. y varies directly as x, and y 5 when x 3. y 5 3 x 16. m varies directly as w, and m 1 2 when w 1 4. m 2w 17. A varies inversely as B, and A 3 when B 2. 6 A B 18. c varies inversely as d, and c 5 when d 2. c 1 d 19. m varies inversely as p, and m 22 when p 9. m 19 8 p 2. s varies inversely as v, and s 3 when v 4. 12 s v 21. A varies jointly as t and u, and A 24 when t 6 and u 2. A 2tu 22. N varies jointly as p and q, and N 72 when p 3 and q 2. N 12pq 23. T varies directly as u, and T 9 when u 2. 9 T 2 u 24. R varies directly as p, and R 3 when p 6. R 5p Solve each variation problem. See Examples 2 4. 25. Y varies directly as x, and Y 1 when x 2. Find Y when x 5. 25 26. n varies directly as q, and n 39 when q 3. Find n when q 8. 14 27. a varies inversely as b, and a 3, when b 4. Find a when b 12. 1 28. y varies inversely as w, and y 9 when w 2. Find y when w 6. 3 29. P varies jointly as s and t, and P 56 when s 2 and t 4. Find P when s 5 and t 3. 15
196 (3-62) Chapter 3 Linear Equations in Two Variables and Their Graphs 3. B varies jointly as u and v, and B 12 when u 4 and v 6. Find B when u 5 and v 8. 2 Solve each problem. 31. Aluminum flatboat. The weight of an aluminum flatboat varies directly with the length of the boat. If a 12-foot boat weighs 86 pounds, then what is the weight of a 14-foot boat? 1.3 pounds 32. Christmas tree. The price of a Christmas tree varies directly with the height. If a 5-foot tree costs $2, then what is the price of a 6-foot tree? $24 33. Sharing the wor. The time it taes to erect the big circus tent varies inversely as the number of elephants woring on the job. If it taes four elephants 75 minutes, then how long would it tae six elephants? 5 minutes Time (minutes) 1 75 5 25 1 2 3 4 5 6 7 8 9 1 Number of elephants FIGURE FOR EXERCISE 33 34. Gas laws. The volume of a gas is inversely proportional to the pressure on the gas. If the volume is 6 cubic centimeters when the pressure on the gas is 8 ilograms per square centimeter, then what is the volume when the pressure is 12 ilograms per square centimeter? 4 cm 3 35. Steel tubing. The cost of steel tubing is jointly proportional to its length and diameter. If a 1-foot tube with a 1-inch diameter costs $5.8, then what is the cost of a 15-foot tube with a 2-inch diameter? $17.4 36. Sales tax. The amount of sales tax varies jointly with the number of Coes purchased and the price per Coe. If the sales tax on eight Coes at 65 cents each is 26 cents, then what is the sales tax on six Coes at 9 cents each? 27 cents 37. Approach speed. The approach speed of an airplane is directly proportional to its landing speed. If the approach speed for a Piper Cheyenne is 9 mph with a landing speed Approach speed (mph) 12 1 9 of 75 mph, then what is the landing speed for an airplane with an approach speed of 96 mph? 8 mph 11 8 7 6 5 5 6 7 8 9 1 Landing speed (mph) FIGURE FOR EXERCISE 37 38. Ideal waist size. According to Dr. Aaron R. Folsom of the University of Minnesota School of Public Health, your maximum ideal waist size is directly proportional to your hip size. For a woman with 4-inch hips, the maximum ideal waist size is 32 inches. What is the maximum ideal waist size for a woman with 35-inch hips? 28 inches GETTING MORE INVOLVED 39. Discussion. If y varies directly as x, then the graph of the equation is a straight line. What is its slope? What is the y-intercept? If y 3x 2, then does y vary directly as x? Which straight lines correspond to direct variations?, (, ), no, y x 4. Writing. Write a summary of the three types of variation. Include an example of each type that is not found in this text. COLLABORATIVE ACTIVITIES Inches or Centimeters? In this activity you will generate data by measuring in both inches and centimeters the height of each member of your group. Then you will plot the points on a graph and use any two of your points to find the conversion formula for converting inches to centimeters. Grouping: 3 to 4 students Topic: Plotting points, graphing lines Height Height Name in inches in centimeters Part I: Measure the height of each person in your group and fill out a table lie the one shown here: