Combined Variation - Purdue University

[Pages:3]16-week Lesson 37 (8-week Lesson 31)

Combined Variation

Combined Variation: - a combination of direct and indirect variation, or joint and indirect variation o when a quantity varies directly (or jointly) with one or more variables and inversely with one or more variables - described by formulas such as = , where varies directly with and and inversely with o depending on how the variables in the numerator and/or denominator change (increasing or decreasing), the dependent variable could increase, decrease, or remain unchanged

- Example of combined variation:

o Newton's law of universal gravitation the formula is = 122, where is the gravitational force between two objects, 1 is the mass of one object, 2 is the mass of another object, is the distance between the two objects, and is the constant of variation

for this formula, the constant of variation is the

gravitational constant (0.000000000066743...)

Example 1: Express the following statement as a formula that involves the given variables and a constant of proportionality , and then determine the value of from the given conditions.

varies directly as and indirectly as . If = 2 and = 4, then = 7.

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16-week Lesson 37 (8-week Lesson 31)

Combined Variation

Example 2: Express the following statement as a formula that involves the given variables and a constant of proportionality , and then determine the value of from the given conditions.

is jointly proportional to the square root of and the cube of , and inversely proportional to the 5.

If = 9, = 5, and = 2, then = 17.

Example 3: The centrifugal force of a body moving in a circle varies jointly with the radius of the circular path and the body's mass , and inversely with the square of the time it takes to move about one full circle.

a. Express the statement above as a formula.

b. A 6-gram body moving in a circle with radius 100 centimeters at a rate of 1 revolution every 2 seconds has a centrifugal force of 6,000 dynes. Use this information to determine the value of .

c. Find the centrifugal force of an 18-gram body moving in a circle with radius 100 centimeters at a rate of 1 revolution every 3 seconds?

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16-week Lesson 37 (8-week Lesson 31)

Combined Variation

Example 4: In baseball, a pitcher's earned-run average varies directly as the number of earned runs allowed and inversely as the number of innings pitched .

a. Express the previous statement as a formula.

b. If a pitcher has an earned-run average of 3.6 after pitching 95 innings and allowing 38 earned-runs, what is the value of ?

(38) 3.6 = 95 95(3.6) = 38

95(3.6) 38 = = 9

c. What is the earned-run average of a pitcher who gave up 69 earned runs in 308 innings? Round to the hundredths place.

9(69) = 308

621 = 308 2.02

Answers to Examples:

1.

=

;

=

14

;

2.

=

3 5

;

=

544 375

;

3a.

=

2

;

3b.

= 40 ; 3c.

= 8,000 ;

4a.

=

;

4b.

= 9 ; 4c.

= 2.02 ;

3

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