6.7 Variation and Problem Solving. OBJECTIVES 1 Solve Problems Involving Direct Variation. 2 Solve Problems Involving Inverse Variation.

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1 390 CHAPTER 6 Rational Epressions 66. A doctor recorded a body-mass inde of 7 on a patient s chart. Later, a nurse notices that the doctor recorded the patient s weight as 0 pounds but neglected to record the patient s height. Eplain how the nurse can use the information from the chart to find the patient s height. Then find the height. In physics, when the source of a sound is traveling toward an observer, the relationship between the actual pitch a of the sound and the pitch h that the observer hears due to the Doppler effect is described by the formula h = a - s, where s is the speed of the 770 sound source in miles per hour. Use this formula to answer Eercise 67 and An emergency vehicle has a single-tone siren with the pitch of the musical note E. As it approaches an observer standing by the road, the vehicle is traveling 50 mph. Is the pitch that the observer hears due to the Doppler effect lower or higher than the actual pitch? To which musical note is the pitch that the observer hears closest? Pitch of an Octave of Musical Notes in Hertz (Hz) Note Pitch Middle C 6.63 D E F 39.3 G A 0.00 B Suppose an emergency van has a single-tone siren with the pitch of the musical note G. If the van is traveling at 80 mph approaching a standing observer, name the pitch the observer hears (rounded to the nearest tenth) and the musical note closest to that pitch. In electronics, the relationship among the resistances R and R of two resistors wired in a parallel circuit and their combined resistance R is described by the formula R = R + R. Use this formula to solve Eercises 69 through If the combined resistance is ohms and one of the two resistances is 3 ohms, find the other resistance. 70. Find the combined resistance of two resistors of ohms each when they are wired in a parallel circuit. 7. The relationship among resistance of two resistors wired in a parallel circuit and their combined resistance may be etended to three resistors of resistances R, R, and R 3. Write an equation you believe may describe the relationship and use it to find the combined resistance if R is 5, R is 6, and R 3 is. 7. For the formula = y + z -, find if y =, z = 7, w and w = 6. Note: Greater numbers indicate higher pitches (acoustically). (Source: American Standards Association) 6.7 Variation and Problem Solving S Solve Problems Involving Direct Variation. Solve Problems Involving Inverse Variation. 3 Solve Problems Involving Joint Variation. Solve Problems Involving Combined Variation. Solving Problems Involving Direct Variation A very familiar eample of direct variation is the relationship of the circumference C of a circle to its radius r. The formula C = pr epresses that the circumference is always p times the radius. In other words, C is always a constant multiple p of r. Because it is, we say that C varies directly as r, that C varies directly with r, or that C is directly proportional to r. C pr constant

2 Section 6.7 Variation and Problem Solving 39 Direct Variation y varies directly as, or y is directly proportional to, if there is a nonzero constant k such that y = k The number k is called the constant of variation or the constant of proportionality. In the above definition, the relationship described between and y is a linear one. In other words, the graph of y = k is a line. The slope of the line is k, and the line passes through the origin. For eample, the graph of the direct variation equation C = pr is shown. The horizontal ais represents the radius r, and the vertical ais is the circumference C. From the graph, we can read that when the radius is 6 units, the circumference is approimately 38 units. Also, when the circumference is 5 units, the radius is between 7 and 8 units. Notice that as the radius increases, the circumference increases. C C increases C pr r as r increases EXAMPLE Suppose that y varies directly as. If y is 5 when is 30, find the constant of variation and the direct variation equation. Solution Since y varies directly as, we write y = k. If y = 5 when = 30, we have that y = k 5 = k30 Replace y with 5 and with = k The constant of variation is 6. Solve for k. After finding the constant of variation k, the direct variation equation can be written as y = 6. Suppose that y varies directly as. If y is 0 when is 5, find the constant of variation and the direct variation equation. EXAMPLE Using Direct Variation and Hooke s Law Hooke s law states that the distance a spring stretches is directly proportional to the weight attached to the spring. If a 0-pound weight attached to the spring stretches the spring 5 inches, find the distance that a 65-pound weight attached to the spring stretches the spring. (Continued on net page)

3 39 CHAPTER 6 Rational Epressions Solution. UNDERSTAND. Read and reread the problem. Notice that we are given that the distance a spring stretches is directly proportional to the weight attached. We let d = the distance stretched w = the weight attached The constant of variation is represented by k. d. TRANSLATE. Because d is directly proportional to w, we write 65 lb d = kw 3. SOLVE. When a weight of 0 pounds is attached, the spring stretches 5 inches. That is, when w = 0, d = 5. d = kw 5 = k0 Replace d with 5 and w with 0. 8 = k Solve for k. Now when we replace k with in the equation d = kw, we have 8 d = 8 w To find the stretch when a weight of 65 pounds is attached, we replace w with 65 to find d. d = 8 65 = 65 8 = 8 8 or 8.5. INTERPRET. Check: Check the proposed solution of 8.5 inches in the original problem. State: The spring stetches 8.5 inches when a 65-pound weight is attached. Use Hooke s law as stated in Eample. If a 36-pound weight attached to a spring stretches the spring 9 inches, find the distance that a 75-pound weight attached to the spring stretches the spring. Solving Problems Involving Inverse Variation When y is proportional to the reciprocal of another variable, we say that y varies inversely as, or that y is inversely proportional to. An eample of the inverse variation relationship is the relationship between the pressure that a gas eerts and the volume of its container. As the volume of a container decreases, the pressure of the gas it contains increases. Inverse Variation y varies inversely as, or y is inversely proportional to, if there is a nonzero constant k such that y = k The number k is called the constant of variation or the constant of proportionality.

4 Section 6.7 Variation and Problem Solving 393 Notice that y = k is a rational equation. Its graph for k 7 0 and 7 0 is shown. From the graph, we can see that as increases, y decreases. y y decreases k y, k 0, 0 as increases EXAMPLE 3 Suppose that u varies inversely as w. If u is 3 when w is 5, find the constant of variation and the inverse variation equation. Solution Since u varies inversely as w, we have u = k. We let u = 3 and w = 5, w and we solve for k. u = k w 3 = k 5 Let u = 3 and w = 5. 5 = k Multiply both sides by 5. The constant of variation k is 5. This gives the inverse variation equation u = 5 w 3 Suppose that b varies inversely as a. If b is 5 when a is 9, find the constant of variation and the inverse variation equation. EXAMPLE Using Inverse Variation and Boyle s Law Boyle s law says that if the temperature stays the same, the pressure P of a gas is inversely proportional to the volume V. If a cylinder in a steam engine has a pressure of 960 kilopascals when the volume is. cubic meters, find the pressure when the volume increases to.5 cubic meters. Solution. UNDERSTAND. Read and reread the problem. Notice that we are given that the pressure of a gas is inversely proportional to the volume. We will let P = the pressure and V = the volume. The constant of variation is represented by k.. TRANSLATE. Because P is inversely proportional to V, we write P = k V When P = 960 kilopascals, the volume V =. cubic meters. We use this information to find k. 960 = k Let P = 960 and V =... 3 = k Multiply both sides by.. Thus, the value of k is 3. Replacing k with 3 in the variation equation, we have P = 3 V Net we find P when V is.5 cubic meters. (Continued on net page)

5 39 CHAPTER 6 Rational Epressions 3. SOLVE. P = 3.5 = Let V =.5.. INTERPRET. Check: Check the proposed solution in the original problem. State: When the volume is.5 cubic meters, the pressure is kilopascals. Use Boyle s law as stated in Eample. When P = 350 kilopascals and V =.8 cubic meters, find the pressure when the volume decreases to.5 cubic meters. 3 Solving Problems Involving Joint Variation Sometimes the ratio of a variable to the product of many other variables is constant. For eample, the ratio of distance traveled to the product of speed and time traveled is always. d = or d = rt rt Such a relationship is called joint variation. Joint Variation If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional to, the other variables. If y = kz then the number k is the constant of variation or the constant of proportionality. CONCEPT CHECK Which type of variation is represented by the equation y = 8? Eplain. a. Direct variation b. Inverse variation c. Joint variation EXAMPLE 5 Epressing Surface Area The lateral surface area of a cylinder varies jointly as its radius and height. Epress this surface area S in terms of radius r and height h. r h Solution Because the surface area varies jointly as the radius r and the height h, we equate S to a constant multiple of r and h. S = krh Answer to Concept Check: b; answers may vary In the equation, S = krh, it can be determined that the constant k is p, and we then have the formula S = prh. (The lateral surface area formula does not include the areas of the two circular bases.)

6 Section 6.7 Variation and Problem Solving The area of a regular polygon varies jointly as its apothem and its perimeter. Epress the area in terms of the apothem a and the perimeter p. Solving Problems Involving Combined Variation Some eamples of variation involve combinations of direct, inverse, and joint variation. We will call these variations combined variation. EXAMPLE 6 Suppose that y varies directly as the square of. If y is when is, find the constant of variation and the variation equation. Solution Since y varies directly as the square of, we have y = k Now let y = and = and solve for k. y = k = k # = k 6 = k The constant of variation is 6, so the variation equation is y = 6 6 Suppose that y varies inversely as the cube of. If y is when is, find the constant of variation and the variation equation. EXAMPLE 7 Finding Column Weight The maimum weight that a circular column can support is directly proportional to the fourth power of its diameter and is inversely proportional to the square of its height. A -meter-diameter column that is 8 meters in height can support ton. Find the weight that a -meter-diameter column that is meters in height can support. ton 8 m? m m m Solution. UNDERSTAND. Read and reread the problem. Let w = weight, d = diameter, h = height, and k = the constant of variation.. TRANSLATE. Since w is directly proportional to d and inversely proportional to h, we have w = kd h (Continued on net page)

7 396 CHAPTER 6 Rational Epressions 3. SOLVE. To find k, we are given that a -meter-diameter column that is 8 meters in height can support ton. That is, w = when d = and h = 8, or = k # 8 = k # 6 6 = k Let w =, d =, and h = 8. Solve for k. Now replace k with in the equation w = kd and we have h w = d h To find weight w for a -meter-diameter column that is meters in height, let d = and h =. w = # w = 6 =. INTERPRET. Check: Check the proposed solution in the original problem. State: The -meter-diameter column that is meters in height can support ton of weight. 7 Suppose that y varies directly as z and inversely as the cube of. If y is 5 when z = 5 and = 3, find the constant of variation and the variation equation. Vocabulary, Readiness & Video Check State whether each equation represents direct, inverse, or joint variation.. y = 5. y = y = y = 5z. y = abc 6. y =.3 7. y = 8. y = 3.st 3 Martin-Gay Interactive Videos See Video 6.7 Watch the section lecture video and answer the following questions Based on the lecture before Eample, what kind of equation is a direct variation equation? What does k, the constant of variation, represent in this equation? 0. In Eample 3, why is it not necessary to replace the given values of and y in the inverse variation equation in order to find k?. Based on Eample 5 and the lecture before, what is the variation equation for y varies jointly as the square of a and the fifth power of b?. From Eample 6, what kind of variation does a combined variation application involve?

8 Section 6.7 Variation and Problem Solving Eercise Set If y varies directly as, find the constant of variation and the direct variation equation for each situation. See Eample.. y = when = 0. y = 5 when = y = 6 when =. y = when = 8 5. y = 7 when = 6. y = when = 3 7. y = 0. when = y = 0. when =.5 Solve. See Eample. 9. The weight of a synthetic ball varies directly with the cube of its radius. A ball with a radius of inches weighs.0 pounds. Find the weight of a ball of the same material with a 3-inch radius. Solve. See Eample.. Pairs of markings a set distance apart are made on highways so that police can detect drivers eceeding the speed limit. Over a fied distance, the speed R varies inversely with the time T. In one particular pair of markings, R is 5 mph when T is 6 seconds. Find the speed of a car that travels the given distance in 5 seconds.. The weight of an object on or above the surface of Earth varies inversely as the square of the distance between the object and Earth s center. If a person weighs 60 pounds on Earth s surface, find the individual s weight if he moves 00 miles above Earth. Round to the nearest whole pound. (Assume that Earth s radius is 000 miles.)? pounds 00 miles 60 pounds 0. At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer. If a person who is 36 feet above the water can see 7. miles, find how far a person 6 feet above the water can see. Round to the nearest tenth of a mile.. The amount P of pollution varies directly with the population N of people. Kansas City has a population of 60,000 and produces about 70,000 tons of pollutants. Find how many tons of pollution we should epect St. Louis to produce if we know that its population is 39,000. Round to the nearest whole ton. (Source: Wikipedia). Charles s law states that if the pressure P stays the same, the volume V of a gas is directly proportional to its temperature T. If a balloon is filled with 0 cubic meters of a gas at a temperature of 300 K, find the new volume if the temperature rises to 360 K while the pressure stays the same. If y varies inversely as, find the constant of variation and the inverse variation equation for each situation. See Eample y = 6 when = 5. y = 0 when = 9 5. y = 00 when = 7 6. y = 63 when = 3 r 3. If the voltage V in an electric circuit is held constant, the current I is inversely proportional to the resistance R. If the current is 0 amperes when the resistance is 70 ohms, find the current when the resistance is 50 ohms.. Because it is more efficient to produce larger numbers of items, the cost of producing a certain computer DVD is inversely proportional to the number produced. If 000 can be produced at a cost of $.0 each, find the cost per DVD when 6000 are produced. 5. The intensity I of light varies inversely as the square of the distance d from the light source. If the distance from the light source is doubled (see the figure), determine what happens to the intensity of light at the new location. 7. y = when = y = when = y = 0. when = y = 0.6 when = 0.3 in. in.

9 398 CHAPTER 6 Rational Epressions 6. The maimum weight that a circular column can hold is inversely proportional to the square of its height. If an 8-foot column can hold tons, find how much weight a 0-foot column can hold.?. The volume of a cone varies jointly as its height and the square of its radius. If the volume of a cone is 3p cubic inches when the radius is inches and the height is 6 inches, find the volume of a cone when the radius is 3 inches and the height is 5 inches. r ton 0 feet 8 feet h MIXED Write each statement as an equation. Use k as the constant of variation. See Eample varies jointly as y and z. 8. P varies jointly as R and the square of S. 9. r varies jointly as s and the cube of t. 30. a varies jointly as b and c. For each statement, find the constant of variation and the variation equation. See Eamples 5 and y varies directly as the cube of ; y = 9 when = 3 3. y varies directly as the cube of ; y = 3 when = 33. y varies directly as the square root of ; y = 0. when = 3. y varies directly as the square root of ; y =. when = y varies inversely as the square of ; y = 0.05 when = y varies inversely as the square of ; y = 0.0 when = y varies jointly as and the cube of z; y = 0 when = 5 and z = 38. y varies jointly as and the square of z; y = 360 when = and z = 3 Solve. See Eample The maimum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as its length. If a beam foot wide, foot high, and 3 0 feet long can support tons, find how much a similar beam can support if the beam is 3 foot wide, foot high, and 6 feet long. 0. The number of cars manufactured on an assembly line at a General Motors plant varies jointly as the number of workers and the time they work. If 00 workers can produce 60 cars in hours, find how many cars 0 workers should be able to make in 3 hours.. When a wind blows perpendicularly against a flat surface, its force is jointly proportional to the surface area and the speed of the wind. A sail whose surface area is square feet eperiences a 0-pound force when the wind speed is 0 miles per hour. Find the force on an 8-square-foot sail if the wind speed is miles per hour. 3. The intensity of light (in foot-candles) varies inversely as the square of, the distance in feet from the light source. The intensity of light feet from the source is 80 foot-candles. How far away is the source if the intensity of light is 5 footcandles?. The horsepower that can be safely transmitted to a shaft varies jointly as the shaft s angular speed of rotation (in revolutions per minute) and the cube of its diameter. A -inch shaft making 0 revolutions per minute safely transmits 0 horsepower. Find how much horsepower can be safely transmitted by a 3-inch shaft making 80 revolutions per minute. MIXED Write an equation to describe each variation. Use k for the constant of proportionality. See Eamples through y varies directly as 6. p varies directly as q 7. a varies inversely as b 8. y varies inversely as 9. y varies jointly as and z 50. y varies jointly as q, r, and t 5. y varies inversely as 3 5. y varies inversely as a 53. y varies directly as and inversely as p 5. y varies directly as a 5 and inversely as b REVIEW AND PREVIEW Find the eact circumference and area of each circle. See the inside cover for a list of geometric formulas in. 6 cm

10 Chapter 6 Vocabulary Check cm 58. Find each square root. See Section A A A 9 CONCEPT EXTENSIONS A Solve. See the Concept Check in this section. Choose the type of variation that each equation represents. a. Direct variation b. Inverse variation c. Joint variation 67. y = y = y = 9ab 70. y = 7 m 7. The horsepower to drive a boat varies directly as the cube of the speed of the boat. If the speed of the boat is to double, determine the corresponding increase in horsepower required. 7. The volume of a cylinder varies jointly as the height and the square of the radius. If the height is halved and the radius is doubled, determine what happens to the volume. 73. Suppose that y varies directly as. If is doubled, what is the effect on y? 7. Suppose that y varies directly as. If is doubled, what is the effect on y? Complete the following table for the inverse variation y = k over each given value of k. Plot the points on a rectangular coordinate system. y = k 75. k = k = 77. k = 78. k = 5 Chapter 6 Vocabulary Check Fill in each blank with one of the words or phrases listed below. rational epression equation comple fraction opposites synthetic division least common denominator epression long division jointly directly inversely. A rational epression whose numerator, denominator, or both contain one or more rational epressions is called a(n).. To divide a polynomial by a polynomial other than a monomial, we use. 3. In the equation y = k, y varies as.. In the equation y = k, y varies as. 5. The of a list of rational epressions is a polynomial of least degree whose factors include the denominator factors in the list. 6. When a polynomial is to be divided by a binomial of the form - c, a shortcut process called may be used. 7. In the equation y = kz, y varies as and z. 8. The epressions - 5 and 5 - are called. 9. A(n) is an epression that can be written as the quotient P of two polynomials P and Q as long as Q Q is not Which is an epression and which is an equation? An eample of an is + = 7, and an eample of an is + 5.

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