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§ 6.8

§ 6.8. Modeling Using Variation. Variation. Certain situations occur so frequently in applied situations that they are given special names. Variation formulas show how one quantity changes in relation to other quantities. Quantities can vary directly, inversely, or jointly.

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§ 6.8

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  1. §6.8 Modeling Using Variation

  2. Variation Certain situations occur so frequently in applied situations that they are given special names. Variation formulas show how one quantity changes in relation to other quantities. Quantities can vary directly, inversely, or jointly. y varies directly as x varying directly as a power of x y varies inversely as x y varies jointly as x and z Blitzer, Intermediate Algebra, 5e – Slide #2 Section 6.8

  3. Direct Variation Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.8

  4. Direct Variation EXAMPLE Use the 4-step procedure for solving the variation problem. y varies directly as x. y = 45 when x = 5. Find y when x = 13. (For example: Purchase 5 pounds of steak for $45.) SOLUTION • Write an equation. We know that y varies directly as x is expressed as y = kx. 2) Use the given values to find k. y = kx Given equation. Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.8

  5. Direct Variation CONTINUED 45 = k(5) Replace y with 45 and x with 5. 9 = k Divide both sides by 9. 3) Substitute the value of k into the equation. For example: $9 a pound. y = 9x Replace k from the equation from step 1 with 9. 4) Answer the problem’s question. For example: How much will it cost for 13 pounds? y = 9(13) Replace x from the equation from step 3 with 13. Will cost $117. y = 117 Multiply. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.8

  6. Direct Variation EXAMPLE An object’s weight on the moon, M,varies directly as its weight on Earth, E. Neil Armstrong, the first person to step on the moon on July 20, 1969, weighed 360 pounds on Earth (with all of his equipment on) and 60 pounds on the moon. What is the moon weight of a person who weighs 186 pounds on Earth? SOLUTION • Write an equation. We know that y varies directly as x is expressed as y = kx. By changing letters, we can write an equation that describes the following English statement: Moon weight, M, varies directly as Earth weight, E. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.8

  7. Direct Variation CONTINUED M = kE. 2) Use the given values to find k. We are told that an object with an earth weight of 360 pounds has a moon weight of 60 pounds. Substitute 360 for E and 60 for M in the direct variation equation. Then solve for k. M = kE Direct variation equation. 60 = k360 Replace M with 60 and E with 360. 1/6 = k Divide both sides by 360. 3) Substitute the value of k into the equation. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.8

  8. Direct Variation CONTINUED M = kE Direct variation equation. M = (1/6)E Replace k with 1/6. 4) Answer the problem’s question. How much does an object with a 186 pound earth weight weigh on the moon? Substitute 186 for E in the preceding equation and then solve for M. M = (1/6)E Equation from step 3. M = (1/6)(186) Replace E with 186. M = 31 Multiply. An object that weighs 186 pounds on the earth weighs 31 pounds on the moon. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.8

  9. Direct Variation Check Point 1 on page 465 The number of gallons of water, W used when taking a shower varies directly as the time, t, in minutes, in the shower. A shower lasting 5 minutes uses 30 gallons of water. How much water is used in a shower lasting 11 minutes? SOLUTION • Write an equation. We know that y varies directly as x is expressed as y = kx. W = kt. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.8

  10. Direct Variation CONTINUED 2) Use the given values to find k. A shower lasting 5 minutes uses 30 gallons of water. Use the given values to find k. W = kt Direct variation equation. 30 = k5 Replace W with 30 and t with 5. 6 = k Divide both sides by 5. 3) Substitute the value of k into the equation. W = 6t Replace k with 6. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.8

  11. Direct Variation CONTINUED 4) Answer the problem’s question. Find W when t is 11 minutes. W = 6t Equation from step 3. W = (6)(11) Replace t with 11. W = 66 Multiply. An 11 minute shower will use 66 gallons of water. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.8

  12. Direct Variation With Powers Next, we consider y varying directly as a power of x: Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.8

  13. Direct Variation EXAMPLE On a dry asphalt road, a car’s stopping distance varies directly as the square of its speed. A car traveling at 45 miles per hour can stop in 67.5 feet. What is the stopping distance for a car traveling at 60 miles per hour? SOLUTION • Write an equation. We know that y varies directly as the square of x is expressed as By changing letters, we can write an equation that describes the following English statement: Distance, d, varies directly as the square of the speed, s. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.8

  14. Direct Variation CONTINUED 2) Use the given values to find k. We are told that a car traveling at 45 miles per hour can stop in 67.5 feet. Substitute 45 for s and 67.5 for d. Then solve for k. This is the equation from step 1. Replace d with 67.5 and s with 45. Simplify. Divide both sides by 2025. 3) Substitute the value of k into the equation. This is the equation from step 1. Replace k with 0.033. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.8

  15. Direct Variation CONTINUED 4) Answer the problem’s question. What is the stopping distance for a car traveling at 60 miles per hour? Substitute 60 for s in the preceding equation. This is the equation from step 3. Replace s with 60. Simplify. Simplify. The stopping distance for a car traveling at 60 miles per hour is 120 feet. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 6.8

  16. Direct Variation Check Point 2 on page 467 The distance required to stop a car varies directly as the square of its speed. If it requires 200 feet to stop a car traveling 60 miles per hour, how many feet are required to stop a car traveling 100 miles per hour? SOLUTION • Write an equation. By changing letters, we can write an equation that describes the following English statement: Stopping distance, s, varies directly as the square of the speed of the car, v. 2) Use the given values to find k. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 6.8

  17. Direct Variation CONTINUED 3) Substitute the value of k into the equation. This is the equation from step 1. 4) Answer the problem’s question: find s when v=100. Replace v with 100. Simplify. About 556 feet are required to stop a car traveling 100 miles per hour. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 6.8

  18. Inverse Variation And now we consider inverse variation… Blitzer, Intermediate Algebra, 5e – Slide #18 Section 6.8

  19. Inverse Variation EXAMPLE The water temperature of the Pacific Ocean varies inversely as the water’s depth. At a depth of 1000 meters, the water temperature is Celsius. What is the water temperature at a depth of 5000 meters? SOLUTION • Write an equation. We know that y varies inversely as x is expressed as By changing letters, we can write an equation that describes the following English statement: Temperature, T, varies inversely as the water’s depth, D. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 6.8

  20. Inverse Variation CONTINUED 2) Use the given values to find k. The temperature of water in the Pacific Ocean at a depth of 1000 meters is Celsius. Substitute 1000 for D and 4.4 for T in the inverse variation equation. Then solve for k. This is the equation from step 1. Replace T with 4.4 and D with 1000. Multiply both sides by 1000. Blitzer, Intermediate Algebra, 5e – Slide #20 Section 6.8

  21. Inverse Variation CONTINUED 3) Substitute the value of k into the equation. This is the equation from step 1. Replace k with 4400. 4) Answer the problem’s question. We need to find the temperature of the water at a depth of 5000 meters. Substitute 5000 for D. This is the equation from step 3. Blitzer, Intermediate Algebra, 5e – Slide #21 Section 6.8

  22. Inverse Variation CONTINUED Replace D with 5000. Simplify. The temperature of the Pacific Ocean at a depth of 5000 meters is Celsius. Blitzer, Intermediate Algebra, 5e – Slide #22 Section 6.8

  23. Inverse Variation Check Point 3 on page 468 The length of a violin string varies inversely as the frequency of its vibrations. A violin string 8 inches long vibrates at a frequency of 640 cycles per second. What is the frequency of a 10-inch string? SOLUTION • Write an equation. We know that y varies inversely as x is expressed as By changing letters, we can write an equation that describes the following English statement: Frequency, f. varies inversely as the water’s depth, D. 2) Use the given values to find k. Blitzer, Intermediate Algebra, 5e – Slide #23 Section 6.8

  24. Inverse Variation CONTINUED 3) Substitute the value of k into the equation. This is the equation from step 1. 4) Answer the problem’s question: find f when l=10. Replace v with 100. Simplify. A string length of 10 inches will vibrate at 512 cycles per second. Blitzer, Intermediate Algebra, 5e – Slide #24 Section 6.8

  25. Combined Variation In combined variation, direct and inverse variation occur at the same time. For example, as the advertising budget, A, of a company increases, its monthly Sales, S, also increase. Then S increases directly as A and S = kA But by contrast, as the price of the company’s product P increases, the company’s monthly sales decrease. Therefore, monthly sales S vary inversely as the price P of the product. That is, We can combine these two variation equations into one equation: This is an example of combined variation. Blitzer, Intermediate Algebra, 5e – Slide #25 Section 6.8

  26. Direct & Inverse (Combined) Variation EXAMPLE One’s intelligence quotient, or IQ, varies directly as a person’s mental age and inversely as that person’s chronological age. A person with a mental age of 25 and a chronological age of 20 has an IQ of 125. What is the chronological age of a person with a mental age of 40 and an IQ of 80? SOLUTION 1) Write an equation. We know that y varies directly as x and inversely as z is expressed as By changing letters, we can write an equation that describes the following English statement: IQ, I, varies directly as the mental age, M, and inversely as chronological age, C. Blitzer, Intermediate Algebra, 5e – Slide #26 Section 6.8

  27. Direct & Inverse (Combined) Variation CONTINUED 2) Use the given values to find k. A person with a mental age of 25 and a chronological age of 20 has an IQ of 125. Substitute 125 for I, 25 for M, and 20 for C. This is the equation from step 1. Replace I with 125, M with 25, and C with 20. Multiply both sides by 20. Divide both sides by 25. Blitzer, Intermediate Algebra, 5e – Slide #27 Section 6.8

  28. Direct & Inverse (Combined) Variation CONTINUED 3) Substitute the value of k into the equation. This is the equation from step 1. Replace k with 100. 4) Answer the problem’s question. We need to determine the chronological age of a person who has a mental age of 40 and an IQ of 80. Substitute 40 for M and 80 for I. This is the equation from step 3. Replace M with 40 and I with 80. Blitzer, Intermediate Algebra, 5e – Slide #28 Section 6.8

  29. Direct & Inverse (Combined) Variation CONTINUED Multiply. Multiply both sides by C. Divide both sides by 80. When a person has a mental age of 40 and an IQ of 80, his/her chronological age is 50 years old. Blitzer, Intermediate Algebra, 5e – Slide #29 Section 6.8

  30. Direct & Inverse (Combined) Variation Check Point 4 on page 470 The number of minutes needed to solve an exercise set of variation problems varies directly as the number of problems and inversely as the number of people working to solve the problems. It takes 4 people 32 minutes to solve 16 problems. How many minutes will it tak 8 people to solve 24 problems? SOLUTION 1) Write an equation.y varies directly as x and inversely as z m=the number of minutes needed to solve an exercise set (32)p =the number of people working on the problems (4)x= the number if problems in the exercise set (16) 2) Use the given values to find k. Blitzer, Intermediate Algebra, 5e – Slide #30 Section 6.8

  31. Direct & Inverse (Combined) Variation CONTINUED 3) Substitute the value of k into the equation. This is the equation from step 1. Replace k with 8 4) Answer the problem’s question. Find m when p=8 and x = 24 Replace M with 40 and I with 80. It will take 24 minutes for 8 people to solve 24 problems. Blitzer, Intermediate Algebra, 5e – Slide #31 Section 6.8

  32. Joint Variation Finally, we have joint variation. Blitzer, Intermediate Algebra, 5e – Slide #32 Section 6.8

  33. Joint Variation EXAMPLE Kinetic energy varies jointly as the mass and the square of the velocity. A mass of 8 grams and velocity of 3 centimeters per second has a kinetic energy of 36 ergs. Find the kinetic energy for a mass of 4 grams and velocity of 6 centimeters per second. SOLUTION Translate “Kinetic energy, E, varies jointly as the mass, M, and the square of the velocity, V.” If M = 8, V = 3, then E = 36. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #33 Section 6.8

  34. Joint Variation CONTINUED Divide both sides by 72. Substitute 0.5 for k in the model for kinetic energy. Find E when M = 4 and V = 6. Simplify. The kinetic energy is 72 ergs. Blitzer, Intermediate Algebra, 5e – Slide #34 Section 6.8

  35. Joint Variation Check Point on page 471 The volume of a cone, V, varies jointly as its height, h, ad the square of its radius r. A cone with a radius measuring a6 feet and a height measuring 10 feet has a volume of 120pi cubic feet. Find the volume of a cone having a radius of 12 feet and a height of 2 feet. SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #35 Section 6.8

  36. DONE

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