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6.5

Variation

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Direct Variation

Variation is an equation that relates one variable to one or more other variables.

In direct variation, the values of the two related variables increase or decrease together.

If a variable y varies directly with a variable x, then

y = kxwhere k is the constant of proportionality (or the variation constant).

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Example

Write the variation and determine the quantity indicated

y varies directly as x. Determine y when x = 15 and k = 8

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Try this

Write the variation and determine the quantity indicated

m varies directly as the square of n. Determine m when n = 3 and k = -4

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Example

The amount of interest earned on an investment, I, varies directly as the interest rate, r. If the interest earned is $50 when the interest rate is 5%, find the amount of interest earned when the interest rate is 7%.

I = rk $50 = 0.05k1000 = k

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Example continued

k = 1000, r = 7%I = rkI = 0.07(1000)I = $70

The amount of interest earned is $70.

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P. 327 # 41

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Inverse Variation

When two quantities vary inversely, as one quantity increases, the other quantity decreases, and vice versa.

If a variable y varies inversely with a variable, x, then

where k is the constant of proportionality.

y = k

x

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Example

Write the variation and determine the quantity indicated

x varies inversely as y. Determine x when y = 7 and k = 14

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Try this

Write the variation and determine the quantity indicated

x varies inversely as y. Determine x when y = 6 and k = 72

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Example

Suppose y varies inversely as x. If y = 12 when x = 18, find y when x = 21.

Now substitute 216 for k, and find y when x = 21.

1218k

21621

y

kyx

216 k

kyx

10.3y

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Try this

Suppose y varies inversely as x. If y = 8 when x = 15, find y when x = 18

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Joint Variation

One quantity may vary directly as the product of two or more other quantities.

The general form of a joint variation, where y, varies directly as x and z, is

y = kxzwhere k is the constant of proportionality.

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Example

The area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a triangle is 48 in2 when its base is 12 in. and its height is 8 in., find the area of a triangle whose base is 15 in. and whose height is 20 in.

48 (12)(8)k

2150 in.A

A kbh

48 196 2

k

48 (96)k

1(15)(20)2

A

A kbh

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Combined Variation

A varies jointly as B and C and inversely as the square of D. If A = 1 when B = 9, C = 4, and D = 6, find A when B = 8, C = 12, and D = 5.

Write the equation.

2

kBCAD

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Combined Variation continued Find the constant of

proportionality. Now find A.

1 k 3.84A

36136

k

2

(9)(4)16

k

2

kBCAD

9625

A

2

(1)(8)(12)5

A

2

kBCAD

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p. 327 #32

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Homework

P. 326 # 23 – 40, 42, 45, 48

Ch. 6.3-6.4 Quiz next class