inverse, joint, and combined variation objective: to find the constant of variation for many types...
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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
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