Variations: Direct, Inverse, and Joint

Direct Variation: One variable is equal to one term containing a constant number and at least one variable.

y = 3x: linear

y = 4x2: quadratic

y = r2h: more than one variable

3, 4, andare the constant factors, but we don’t always know what the constant is going to be at the beginning, so we designate it as “k”.

y = kx, y = kx2, y = kr2h.

Wording of direct variation:

y = r2; y varies directly with the square of r, or y is proportional to the square of r.

Wording of inverse variation:

L = 40/w; L varies inversely with w.

Joint variation:

A direct variation with two or more variations multiplied together.

V = LWH; V varies jointly with L, W, and H

Other possibilities include:

y varies as the sum of x and y; y =k(x+y)

Change the background color to white and then highlight the space to the right of each direction to see the answer.

Examples:

1) y varies directly as the square of x, and y = 8 when x = 2, find y when x = 1.

Direct variation:y = kx2

Given: x=2, y=88 = k(2)2

Find k:8 = k(4) k = 2

Write equation:y = 2x2

Given: x = 1, find yy = 2(1)2

y = 2

2) y varies jointly as x and z, and y = 5 when x = 3, and z = 4, find y when x = 2, and z = 3.

Direct variation:y = kxz

Given: x = 3, z = 4, y = 55 = k(3)(4)

Find k:5 = 12k

k = 5/12

Write equation:y = (5/12)xz

Given: x = 2, z = 3, find yy = (5/12)(2)(3)

y = 5/2

3) y varies inversely with x, and y = 2.5 and when x = 0.4, find y when x = 4.

Inverse variation:y = k/x

Given: y = 2.5, x = 0.42.5 = k/0.4

Find k:k = 1

Write equation:y = 1/x

Given: x = 4, find yy = ¼

Most problems have to be read and interpreted as to what goes where.

4) According to Hooke’s Law, the force needed to stretch a spring is proportional to the distance the spring is stretched. If 50 pounds of force stretches a spring 5 inches, how much will the spring be stretched by a force of 120 pounds?

Direct variation:F = kd

Given: F = 50 lbs., d = 5 50 = k(5)

Find k:k = 10

Write equation:F = 10d

Given:F = 120, find d120 = 10d

distance = 12 inches

5) Kepler’s third law of planetary motion states that the square of the time required for a planet to make one revolution about the sun varies directly as the cube of the average distance of the planet from the sun. If you assume Mars is 1.5 times as far from the sun as is the earth, find the approximate length of a Martian year.

Direct variation:t2 = kd3

Given: earth is 1 AU12 = k(1)3

(Astronomical unit)

in 1 year of time

Find k:k = 1

Write equation:t2 = 1d3

Given d = 1.5, find t:t2 = (1.5)3

t2 = 3.375

t = 1.837

1.837 earth years

6) The weight of a body varies inversely as the square of its distance from the center of the earth. If the radius of the earth is 4000 miles, how much would a 200 pound man weigh 1000 miles above the surface of the earth?

Inverse variation:F = k/d2

Given: d = 4000, F = 200200 = k/(4000)2

Find k:k = 200(4000)2

k = 3,200,000,000

Write equation:F = 3200000000/d2

Given d = 5000, find FF = 3200000000/(5000)2

Weight is 128 lbs.

7) The number of hours it takes men to assemble machines varies directly as x number of machines and inversely as the number of men. If four men can assemble 12 machines in four hours, how many men are needed to assemble 36 machines in eight hours?

Variation:h = kx/m

Given: h = 4, x = 12, m=44 = k(12)/4

Find k:k = 4/3

Write equation:h = (4/3)x/m

Given: h = 8, x = 368 = (4/3)(36)/m

m=(4/3)(36)/8 6 men are needed