Direct Variation. COMPUTERS Use the graph at the right that shows the output of a color printer.


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1 95 Direct Variation MAIN IDEA Use direct variation to solve problems. New Vocabular direct variation constant of variation Math nline glencoe.com Etra Eamples Personal Tutor SelfCheck Quiz CMPUTERS Use the graph at the right that shows the output of a color printer.. What is the constant rate of change, or slope, of the line?. Is the total number of pages printed alwas proportional to the printing time? If so, what is the constant ratio? 3. Compare the constant rate of change to the constant ratio. Pages 8 8 Time (minutes) In the eample above, the number of minutes and the number of pages printed both var, while the ratio of pages printed to minutes,.5 pages per minute, remains constant. When the ratio of two variable quantities is constant, their relationship is called a direct variation. The constant ratio is called the constant of variation. Find a Constant Ratio FUNDRAISER The amount of mone Robin has raised for a bikeathon is shown in the graph at the right. Determine the amount that Robin raises for each mile she rides. Since the graph of the data forms a line, the rate of change is constant. Use the graph to find the constant ratio. Amount Raised ($) 3 8 Distance (miles) amount raised distance _ _ Robin raises $7.5 for each mile she rides. _ _ a. SKYDIVING Two minutes after a skdiver opens his parachute, he has descended,9 feet. After 5 minutes, he has descended,75 feet. If the distance varies directl as the time, at what rate is the skdiver descending? Lesson 95 Direct Variation 87
2 In a direct variation equation, the constant rate of change, or slope, is assigned a special variable, k. Direct Variation Ke Concept Words Smbols Eample A direct variation is a relationship in which the ratio of to is a constant, k. We sa varies directl with. k = _ or = k, where k = 3 Model = 3 Solve a Direct Variation RealWorld Link Most pets age at a different rate than their human companions. For eample, a 3earold dog is often considered to be in human ears. PETS Refer to the information at the left. Assume that the age of a dog varies directl as its equivalent age in human ears. What is the humanear age of a dog that is ears old? Write an equation of direct variation. Let represent the dog s actual age and let represent the humanequivalent age. = k Direct variation = k(3) =, = 3 7 = k Simplif. = 7 Substitute for k = 7. Use the equation to find when =. = 7 = 7() = = Multipl. Proportions In Eample, ou can also use a proportion to solve direct variation problems. Write ratios comparing the human equivalent age to the actual age. 3 = = 3 = A dog that is ears old is ears old in humanequivalent ears. b. SHPPING A grocer store sells oranges for $. How much would it cost to bu oranges? Round to the nearest cent if necessar. In a direct variation, the constant of variation k is a constant rate of change. When the value changes b an amount a, then the value will change b the corresponding amount ka. In the previous eample, when changed b a factor of, changed b 7() or. 88 Chapter 9 Algebra: Linear Functions
3 Look Back To review proportional relationships, see Lessons  and 5. Not all relationships with a constant rate of change are proportional. Likewise, not all linear functions are direct variations. Identif Direct Variation Determine whether each linear function is a direct variation. If so, state the constant of variation. 3 Miles, Gallons, 3 gallons _ miles _ 5 _ 5 5 Compare the ratios to check for a common ratio. 3_ 75 5 _ 5 Since the ratios are the same, the function is a direct variation. The constant of variation is _ 5. Hours, 8 Earnings, earnings _ hours 3_ or _ 8.33 _ The ratios are not the same, so the function is not a direct variation. c. Das, 5 5 Height, d. Time, 8 Distance, Proportional Linear Function Concept Summar Direct Variations Notice that the graph of a direct variation, which is a proportional linear relationship, is a line that passes through the origin. Table Graph Equation _ = Nonproportional Linear Function Table Graph Equation   = _ 5_ 3 3_ Lesson 95 Direct Variation 89
4 Eample (p. 87) Eample (p. 88). MANUFACTURING The number of computers built varies directl as the number of hours the production line operates. What is the ratio of computers built to hours of production?. TRANSPRTATIN A charter bus travels miles in 3 _ hours. Assuming that the distance traveled is directl proportional to the time traveled, how far will the bus travel in hours? Computers Hours Eamples 3, (p. 89) 3. Determine whether the linear function Hours, 3 5 is a direct variation. If so, state the Miles, constant of variation. For Eercises See Eamples 5 5 3, HELP HMEWRK HELP. GARDENING Janelle planted ornamental grass seeds. After the grass breaks the soil surface, its height varies directl with the number of das. What is the rate of growth? Height (in.) 3 5. JBS The amount Dust earns is directl proportional to the number of newspapers he delivers. How much does Dust earn for each newspaper deliver? Earnings ($) Das 8 Newspapers. SUBMARINES Ten minutes after a submarine is launched from a research ship, it is 5 meters below the surface. After 3 minutes, the submarine has descended 75 meters. At what rate is the submarine diving? 7. MVIES The Stratton famil rented 3 DVDs for $.7. The net weekend, the rented 5 DVDs for $7.5. What is the rental fee for a DVD? 8. MEASUREMENT Morgan used 3 gallons of paint to cover,5 square feet and 5 gallons to paint an additional,75 square feet. How man gallons of paint would she need to cover,8 square feet? 9. MEASUREMENT The weight of an object on Mars varies directl with its weight on Earth. An object that weighs 7 pounds on Mars weighs pounds on Earth. If an object weighs pounds on Earth, how much would it weigh on Mars? 9 Chapter 9 Algebra: Linear Functions
5 . ELECTRNICS The height of a widescreen television screen is directl proportional to its width. A manufacturer makes a television screen that is centimeters wide and centimeters high. Find the height of a television screen that is 9 centimeters wide. RealWorld Link The aspect ratio of a television screen describes the ratio of the width of the screen to the height. Standard screens have an aspect ratio of :3 while widescreen televisions have an aspect ratio of :9.. BAKING A cake recipe requires _ 3 cups of flour for servings. How much flour is required to make a cake that serves 3? Determine whether each linear function is a direct variation. If so, state the constant of variation... Pictures, Profit, 8 3 Age, 3 Grade, Minutes, 8 Cost, Price, 5 5 Ta, ALGEBRA If varies directl with, write an equation for the direct variation. Then find each value.. If =  when = 9, find when = Find when = if = 8 when =. 8. If =  when = , what is the value of when = ? 9. Find when = 5, if = 7 when = 8.. Find when = 5, if =. when =.. MEASUREMENT The number of centimeters in a measure varies directl as the number of inches. Find the measure of an object in centimeters if it is 5 inches long. Inches, 9 5 Centimeters, PRACTICE EXTRA PRACTICE See pages 9, 78.. MEASUREMENT The length of the rectangle shown varies directl as its width. What is the perimeter of a rectangle that is meters long? w =. m l = m H..T. Problems 3. PEN ENDED Identif values for and in a direct variation relationship where = 9 when =.. CHALLENGE The amount of stain needed to cover a wood surface is directl proportional to the area of the surface. If 3 pints are required to cover a square deck with a side of 7 feet, how man pints of stain are needed to paint a square deck with a side of feet inches? WRITING IN 5. MATH Write a direct variation equation. Then triple the value and eplain how to find the corresponding change in the value. Lesson 95 Direct Variation 9
6 . Students in a science class recorded lengths of a stretched spring, as shown in the table below. Length of Stretched Spring Distance Stretched, (centimeters) Mass, (grams) Which equation best represents the relationship between the distance stretched and the mass of an object on the spring? A = B = C = _ D = _ 7. SHRT RESPNSE Nicole read pages during a 3minute independent reading period. How man pages would she read in 5 minutes? 8. To make fruit punch, Kelli must add 8 ounces of pineapple juice for ever ounces of orange juice. If she uses 3 ounces of orange juice, which proportion can she use to find, the number of ounces of pineapple juice she should add to make the punch? F 8_ = _ 3 H 8_ = _ 3 G _ 8 = _ 3 J _ = _ 8 3 Find the slope of each line. (Lesson 9) JBS The function p = 7.5h describes the relationship between the number of hours h Callie works and the amount she is paid p. Graph the function. Then use our graph to determine how much Callie can epect to earn if she works hours. (Lesson 93) 33. HEALTH Man health authorities recommend that a health diet contains no more than 3% of its Calories from fat. If Jennie consumes,5 Calories each da, what is the maimum number of Calories she should consume from fat? (Lesson 53) PREREQUISITE SKILL Solve each equation. (Lesson 9) a = = d = n t  = 3 9 Chapter 9 Algebra: Linear Functions
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