1. Section 3-4 Direct and Inverse Variation GoalsGoals Study direct variation Study direct variation Study inverse variation Study inverse variation Study joint variation Study joint variation Study combined variation Study combined variation Solve applied variation problems Solve applied variation problems

2. Direct Variation If a car is traveling at a constant rate of 50 miles per hour, then the distance d traveled in t hours is d = 50t.If a car is traveling at a constant rate of 50 miles per hour, then the distance d traveled in t hours is d = 50t. As t gets larger, d also gets larger.As t gets larger, d also gets larger. As t gets smaller, d also gets smaller.As t gets smaller, d also gets smaller. We say that d is directly proportional to t or d varies directly as t.We say that d is directly proportional to t or d varies directly as t. The number 50 is called the constant of variation or the constant of proportionality.The number 50 is called the constant of variation or the constant of proportionality.

3. Direct Variation, cont’d

4. Example 17 Suppose y varies directly as the square of x.Suppose y varies directly as the square of x. Find the constant of variation if y is 16 when x = 2 and use it to write an equation of variation.Find the constant of variation if y is 16 when x = 2 and use it to write an equation of variation.

5. Example 17, cont’d Solution: Since y varies directly as the square of x, the general equation will be:Solution: Since y varies directly as the square of x, the general equation will be: Use the given values of x and y and solve the equation for k.Use the given values of x and y and solve the equation for k.

6. Example 17, cont’d Solution, cont’d: The constant of variation is k = 4.Solution, cont’d: The constant of variation is k = 4. The variation equation is:The variation equation is:

7. Inverse Variation Boyle’s law for the expansion of gas is, where V is the volume of the gas, P is the pressure, and K is a constant.Boyle’s law for the expansion of gas is, where V is the volume of the gas, P is the pressure, and K is a constant. As P gets larger, V gets smaller.As P gets larger, V gets smaller. As P gets smaller, V gets larger.As P gets smaller, V gets larger. We say that V is inversely proportional to P or V varies inversely as P.We say that V is inversely proportional to P or V varies inversely as P.

8. Inverse Variation, cont’d

9. Example 18 Suppose y varies inversely as the square root of x.Suppose y varies inversely as the square root of x. Find the constant of proportionality if y is 15 when x is 9 and use it to write an equation of variation.Find the constant of proportionality if y is 15 when x is 9 and use it to write an equation of variation.

10. Example 18, cont’d Solution: Since y varies inversely as the square root of x, the general equation will be:Solution: Since y varies inversely as the square root of x, the general equation will be:

11. Example 18, cont’d Solution, cont’d: Replace y with 15 and x with 9, and solve for k.Solution, cont’d: Replace y with 15 and x with 9, and solve for k. The equation isThe equation is

12. Example 19 Determine whether the variation between the variables is direct or inverse.Determine whether the variation between the variables is direct or inverse. a)The time traveled at a constant speed and the distance traveled b)The weight of a car and its gas mileage c)The interest rate and the amount of interest earned on a savings account

13. Example 19, cont’d Solution:Solution: a)The time traveled at a constant speed and the distance traveled The longer you travel at a constant speed the more distance you will cover.The longer you travel at a constant speed the more distance you will cover. This is an example of a direct variation.This is an example of a direct variation.

14. Example 19, cont’d Solution, cont’d:Solution, cont’d: b)The weight of a car and its gas mileage In general, the heavier the car, the lower the miles per gallon for that car.In general, the heavier the car, the lower the miles per gallon for that car. This is an example of inverse variation.This is an example of inverse variation.

15. Example 19, cont’d Solution, cont’d:Solution, cont’d: c)The interest rate and the amount of interest earned on a savings account The higher the interest rate, the more interest you will earn on the account.The higher the interest rate, the more interest you will earn on the account. This is an example of direct variation.This is an example of direct variation.

16. Other Types of Variation

17. Example 20 Write the following statements as a variation equation.Write the following statements as a variation equation. a) w varies jointly as y and the cube of x. b) x is directly proportional to y and inversely proportional to z.

18. Example 20, cont’d Solution:Solution: a) w varies jointly as y and the cube of x. Use the definition for joint variation:Use the definition for joint variation:

19. Example 20, cont’d Solution, cont’d:Solution, cont’d: b) x is directly proportional to y and inversely proportional to z. Use the definition for compound variation:Use the definition for compound variation:

20. Example 21 Suppose y is directly proportional to x and inversely proportional to the square root of z.Suppose y is directly proportional to x and inversely proportional to the square root of z. a)Find the constant of variation if y is 4 when x is 8 and z is 36 and write an equation of variation. b)Determine y when x is 5 and z is 16.

21. Example 21, cont’d Solution: Since y is directly proportional to x and inversely proportional to the square root of z, the general equation is:Solution: Since y is directly proportional to x and inversely proportional to the square root of z, the general equation is:

22. Example 21, cont’d Solution, cont’d:Solution, cont’d: a)Use the given values of y, x, and z to solve for k. The equation isThe equation is

23. Example 21, cont’d Solution, cont’d:Solution, cont’d: b)To determine y when x is 5 and z is 16, evaluate the equation found in part a.

24. Solving Applied Problems

25. Example 22 The distance a car travels at a constant speed varies directly as the time it travels.The distance a car travels at a constant speed varies directly as the time it travels. Find the variation formula for the distance traveled by a car that traveled 220 miles in 4 hours at a constant speed.Find the variation formula for the distance traveled by a car that traveled 220 miles in 4 hours at a constant speed. How many miles will the car travel in 7 hours at that same constant speed?How many miles will the car travel in 7 hours at that same constant speed?

26. Example 22, cont’d Solution: Let d = distance traveled and t = time.Solution: Let d = distance traveled and t = time. The general equation is d = kt.The general equation is d = kt. Substitute 220 for d and 4 for t and solve for k.Substitute 220 for d and 4 for t and solve for k. k = 55 k = 55 The equation is d = 55t.The equation is d = 55t.

27. Example 22, cont’d Solution, cont’d: Since the variation equation is d = 55t, we know the car is traveling at a rate of 55 miles per hour.Solution, cont’d: Since the variation equation is d = 55t, we know the car is traveling at a rate of 55 miles per hour. In 7 hours, the car can travel d = 55(7) = 385 miles.In 7 hours, the car can travel d = 55(7) = 385 miles.

28. Example 23 Ohm’s law says that the current, I, in a wire varies directly as the electromotive force, E, and inversely as the resistance, R.Ohm’s law says that the current, I, in a wire varies directly as the electromotive force, E, and inversely as the resistance, R. If I is 11 when E is 110 and R is 10, find I if E is 220 and R is 11.If I is 11 when E is 110 and R is 10, find I if E is 220 and R is 11.

29. Example 23, cont’d Solution: Since I varies directly as E and inversely as R, the general equation is:Solution: Since I varies directly as E and inversely as R, the general equation is: Substitute 11 for I, 110 for E, and 10 for R, and solve for k.Substitute 11 for I, 110 for E, and 10 for R, and solve for k.

30. Example 23, cont’d Solution, cont’d: The variation equation is:Solution, cont’d: The variation equation is: To find I if E is 220 and R is 11, evaluate the equation.To find I if E is 220 and R is 11, evaluate the equation.