Inverse, Joint, and Combined Variation

Objective: To find the constant of variation for many types of problems

and to solve real world problems.

kxy VariationDirect

2;36 kk 21;63 kk 5.12;3.75.3 kk

kxy VariationDirect

kxy VariationDirect

2;36 kk 21;63 kk 5.12;3.75.3 kk

12936 a

48;4329 aa

b12

936

3;10836 bb

Inverse Variation

• Two variables, x and y, have an inverse-variation relationship if there is a nonzero number k such that xy = k, y = k/x. The constant of variation is k.

Example 1

Example 1

Example 1

Try This

• The variable y varies inversely as x, and y = 120 when x = 6.5. Find the constant of variation and write an equation for the relationship. Then, find y when x is 1.5, 4.5, 8, 12.5, and 14.

Try This

• The variable y varies inversely as x, and y = 120 when x = 6.5. Find the constant of variation and write an equation for the relationship. Then, find y when x is 1.5, 4.5, 8, 12.5, and 14.

780

120 5.6

k

k

xy 780

Try This

• The variable y varies inversely as x, and y = 120 when x = 6.5. Find the constant of variation and write an equation for the relationship. Then, find y when x is 1.5, 4.5, 8, 12.5, and 14.

780

120 5.6

k

k

xy 780

Joint Variation

• If y = kxz, then y varies jointly as x and z, and the constant of variation is k.

Example 2

Example 2

Squared Variation

• If , where k is a nonzero constant, then y varies directly as the square of x. Many geometric relationships involve this type of variation, as show in the next example.

2kxy

Example 3

Example 3

Example 3

Try This

• Write the formula for the area A, of a circle whose radius is r. Identify the type of variation and the constant of variation.

• Find the area of the circle when r is 1.5, 2.5, 3.5, 4.5.

Try This

• Write the formula for the area A, of a circle whose radius is r. Identify the type of variation and the constant of variation.

• Find the area of the circle when r is 1.5, 2.5, 3.5, 4.5.

• The constant of variation is .

2rA

Try This

• Write the formula for the area A, of a circle whose radius is r. Identify the type of variation and the constant of variation.

• Find the area of the circle when r is 1.5, 2.5, 3.5, 4.5.

• The constant of variation is .

2rA

Combined Variation

Example 4

Example 4

Example 4

Homework

• Page 486• 13-27 odd