Inverse Variation There are also situations in which, as one variable gets bigger, it causes the second variable to get smaller. This is called inverse variation. Some examples: y = a x y = a x 2 y = a x They share the quality that they all have asymptotes at x = 0 and y = 0, and can all be written as y = a, where a is still the constant of variation. n x page 7
Decide whether each relationship varies directly, inversely, or neither. Decide whether each relationship varies directly, inversely, or neither. If directly or inversely, give the constant of variation (assume you re solving for the letter that comes first in the alphabet). 5. 9 a = b 6. 4.2 = r s 7. a 3 = 1.2 b page 8
Example 1 Given that y varies directly with x, and that when x = 1 3, y = 3, translate, solve 7 for a, and write an equation. Then find the value of x when y = 5 4. x = Example 2 A Volvo needs 25 feet to come to a stop if the brakes are applied at 20 mph. Assume the braking distance is directly proportional to the square of the speed. (a) Translate, solve for a, and write an equation. (b) Find the distance needed to stop this Volvo if it s going 60 mph, and how fast the Volvo was going if it took 122 feet to stop. Example 3 Suppose y varies inversely with the square root of x, and y = 7 when x = 4. Write an equation and then find y when x = 2. y = page 9
Example 4 The number of street-sweeping machines used to clean the streets of Gotham City varies indirectly with the time needed to do the job. If 125 machines take 10 days to finish a job, how many days would 200 street sweepers need? y = Use two data points and asymptotes to draw a graph, with both axes labeled: page 10
Write an equation for the given relationship. 1. y varies inversely with x Joint Variation 2. z varies jointly with x, y and r 3. y varies inversely with the square of x 4. z varies directly with y and inversely with x 5. x varies jointly with t and r and inversely with s Example 5 The pressure exerted on the floor by a person s shoe heel is directly proportional to his or her weight and inversely proportional to the square of the width of his or her heel. Professor Snarff weighs 200 pounds, wears a shoe with a 3-inch wide heel, and exerts a pressure of 24 pounds per square inch (psi) on the floor. Phoebe Small weighs 100 pounds, and wears spike heels, which are ¼ in wide. How much pressure does she exert? Why do you suppose that commercial airlines do not allow their cabin crew to wear spike heels? page 11
Unit 8 Homework 2 (8.2) 7. Determine whether x and y show direct variation, inverse variation or neither. x y 1.5 40 2.5 24 4 15 7.5 8 10 6 x y 12 132 18 198 23 253 29 319 34 374 x y 4 16 5 11 6.2 10 7 9 11 6 x y 4 21 6 14 8 10.5 8.4 10 12 7 page 12
23. y varies jointly as x and z. If y = 5 when x = 3 and z = 4, find y when x = 6 and z = 8. 24. y varies directly as x 2 and inversely as z. If y = 12 when x = 2 and z = 7, find y when x = 3 and z = 9. 25. y varies jointly as x 3 and z 2. If y = 3 when x = 8 and z = 4, find y when x = 27 and z = 6. page 13
Solve each problem. 1. The bending of a beam varies directly as its mass. A beam is bent 20mm by a mass of 40 kg. How much will the beam bend with a mass of 100 kg? 2. The distance needed to stop a car varies directly as the square of its speed. It requires 120 m to stop a car at 70 km/h. What distance is required to stop a car at 80 km/h? 3. Laura has a mass of 60 kg and is sitting 265cm from the fulcrum of a seesaw. Bill has a mass of 50 kg. How far from the fulcrum must he be to balance the seesaw? (Hint: The distance from the fulcrum varies inversely as the mass). page 14
4. In an electric circuit, the current varies inversely as the resistance. The current is 40 amps when the resistance is 12 ohms. Find the current when the resistance is 20 ohms. 5. The number of hours required to do a job varies inversely as the number of people working. It takes 8 hours for 4 people to paint the inside of a house. How long would it take 5 people to do the job? 6. Cheers varied jointly as the number of fans and the square of the jubilation factor. When there were 100 fans and jubilation factor was 4 there were 1000 cheers. How many cheers were there when there were only 10 fans whose jubilation factor is 20? 7. The number of rabbits varied directly as the number of squirrels and inversely as the number of raccoons. When there were 10 rabbits and 40 squirrels there were only 2 raccoons. How many raccoons went with 5 rabbits and 20 squirrels? page 15