VARIATION
151
third law of planetary motion uses a fractional power. The
period
T of a planet’s
orbit around the sun is proportional to the 3/2 power of its distance
R from the
sun,
T = kR
3/2
. Because Kepler’s first law stated that planets circle the sun in
elliptical paths, the semimajor axis provides the measure of distance.
Many geometry formulas can be expressed as direct variation. Since the area
of a cube is
A = 6s
2
, where
A is the surface area and s is the length of an edge,
it follows that
s =
A
6
. The length of an edge varies directly as the square root
of surface area of the cube. The length of the edge varies directly as the cube root
of the volume
V , s =
√
V
3
.
Joint variation occurs when the dependent variable varies directly as the
product of two or more independent variables. Many geometry formulas are in
joint variation. The volume of a cylinder is
V =
1
3
πr
2
h. The volume V varies
jointly as the radius
r squared and the height h. The constant of variation is
1
3
π.
The volume of a rectangular solid having length
L, width W , and height H is
expressed in the formula,
V = LW H . The volume
varies jointly as length,
width, and height. The constant of variation is 1.
Biologists and medical scientists have provided formulas for the surface area
of a human-being’s skin. The DuBois formula relates area in square centimeters
jointly to the 0.425 power of weight in kilograms and the 0.725 power of height
in centimeters,
A = 71.84W
0.425
H
0.725
. The formula estimates the surface area
for the average adult male to be about 1.8 square meters, and for the average
adult female, about 1.6 square meters.
Population biologists use different kinds of variation to express rates of
change. The change in a population undergoing rapid growth (see
Exponential
Growth ) is
c = rP , where c is the change in the number of organisms, P is the
population count before change, and
r is the rate of change. In 1995, Mexico’s
population was 91.1 million people. It was increasing at a rate of 2.0
percent per
year. The change formula for Mexico would be the direct variation formula,
c =
0.02P . Using the formula to predict the change in population for 1995 to 1996
gives,
c = 1.822 million people. The change for the following year would be
based on 92.2 people. If there is a limit to the population of a country, say
M
people, then the change formula would be
c = kP (M − P ). Change in a popu-
lation varies jointly as the current population and the available capacity for peo-
ple. This leads to a more complex pattern of growth. (See
Logistic Functions.)
Inverse variation occurs when the variables are related through a reciprocal.
If you must travel 200 miles at a constant rate, the distance-rate-time formula
says that
200 = rt. Solving for t gives the equation t =
200
r
.
In this equation,
t
varies inversely as
r. The constant of variation is 200. The independent variable
can be a power. For example, the intensity
I of light falling on an object varies
inversely as the
square of the distance
d from the light. The formula is I =
k
d
2
.
(See
Inverse Square Function.)
The law of the lever is an inverse variation. The distance
d from the fulcrum
in feet needed to stabilize the seesaw with a person who weights
w pounds is
d =
k
w
. If Jane weighs 100 pounds and sits 5
feet from the fulcrum, how far away
will Juan, who weighs 150 pounds, have to sit in order to balance Jane? Use
Jane’s data to find the constant of variation
k: 5 = k/100, so k = 500. Now
solve for Juan’s distance:
d = 500/150 = 3.33. Juan would have to sit 3 feet 4
inches from the fulcrum in order to balance Jane. Note that the constant
k was
computed from Jane’s statistics. If she were to change position or be replaced by
someone else, the value of
k would change.
Pulley systems are a series of ropes and wheels that help lift and support
heavy objects by distributing weight in multiple locations. Elevator shafts rely on
pulleys to move the cabin, and movers use pulleys to transport cumbersome or
heavy objects such as pianos into tall buildings. A 100 pound weight can feel like
a 50 pound weight when it is
moved by a two-pulley system, because half the
weight is distributed at the other pulleys. As the number of pulleys in the system
increases, the amount of force needed to move the object decreases proportion-
ally. Therefore a three-pulley system needs a 33.33-pound force to move the 100
pound weight, a four-pulley system needs a 25 pound force to move the 100
pound weight, and so on. The force,
f , needed to move an object, the weight of
the object,
w, the number of pulleys needed in a system, p, are related with the
equation,
f = w/p. If the weight is constant, then the force applied varies in-
versely with the number of pulleys used.
Compound variation combines direct and indirect
variation with two or more
independent variables. The gravitational force between two planets varies
directly as the product of the masses of the planets, and inversely as the square
of the distance between them:
F =
Gm
1
m
2
d
2
, where
F is the force in newtons, G
is a gravitational constant (
6.67 × 10
−11
newton-meters per square kilogram),
r
is the distance in meters between the centers of two planets, and
m
1
and
m
2
are
the mass of each planet in kilograms. The constant of variation would be differ-
ent if measurements are made in different units, such as in feet rather than meters
and pounds rather than kilograms. The formula works if one of the planets is the
earth and the other “planet” is a person high above the earth’s surface. It simpli-
fies to an inverse-variation formula for the
weight of a body above the
earth:
W =
k
d
2
, where
W is the weight above the planet, d is the distance be-
tween the person and the
center of the earth, and
k is a constant. It may seem
strange that both masses have disappeared, but they are handled by the constant.
Consider a 170 pound astronaut who is 9,000 miles above the surface of the
earth. How much does he weigh at that altitude? First write the equation for his
weight at the surface of the earth. Since the radius of the earth is about 4,000
miles,
170 =
k
4,000
2
. Solving for
k yields, k = 2.72 × 10
9
.
The inverse-square
formula is therefore
W =
2,720,000,000
d
2
. Using this formula with the distance
d =
13,000 miles from the center of the earth gives,
W ≈ 16.09. The astronaut would
weigh about 16 pounds.
The deflection
D of a diving board is a function of the weight W of the diver,
the length of the board
L, the elasticity E of the material making up the board,
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