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, and are 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:
Given: x=2, y=8
Find k:
Write equation:
Given: x = 1, find y
y = kx2
8 = k(2)2
8 = k(4)
k=2
y = 2x2
y = 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:
Given: x = 3, z = 4, y = 5
Find k:
y = kxz
5 = k(3)(4)
5 = 12k
k = 5/12
Write equation:
y = (5/12)xz
Given: x = 2, z = 3, find y y = (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:
Given: y = 2.5, x = 0.4
Find k:
Write equation:
Given: x = 4, find y
y = k/x
2.5 = k/0.4
k=1
y = 1/x
y=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:
Given: F = 50 lbs., d = 5
Find k:
Write equation:
Given: F = 120, find d
F = kd
50 = k(5)
k = 10
F = 10d
120 = 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:
Given: earth is 1 AU
(Astronomical unit)
in 1 year of time
Find k:
Write equation:
Given d = 1.5, find t:
t2 = kd3
12 = k(1)3
k=1
t2 = 1d3
t2 = (1.5)3
t2 = 3.375
t = 3.375  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 = 200 200 = k/(4000)2
Find k:
k = 200(4000)2
k = 3,200,000,000
Write equation:
F = 3200000000/d2
Given d = 5000, find F
F=
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:
Given: h = 4, x = 12, m=4
Find k:
Write equation:
Given: h = 8, x = 36
h = kx/m
4 = k(12)/4
k = 4/3
h = (4/3)x/m
8 = (4/3)(36)/m
m=(4/3)(36)/8
6 men are needed