VARIATION
The concept of variation is essential to understanding the algebra of higher mathematics. After all, variation involves
variables!! Consider the formula d  rt . If the rate remains constant while time and distance vary, then as time increases,
so does the distance traveled. So if you are driving at a constant rate of 50 mi/h, then the function d = 50t implies that the
longer you drive, the farther you will go. It also means that the less time spent traveling, the less distance is traveled. This
is an example of direct variation, which is a linear function of the form
y  kx , where k is called the constant of variation
or the constant of proportionality.
DIRECT VARIATION
The statements below are equivalent:
1.
y  kx
INVERSE VARIATION
The statements below are equivalent:
for some constant k
y
1.
2. y varies directly as x
3. y is directly proportional to x.
k
x
for some constant k
2. y varies inversely as x
3. y is inversely proportional to x.
Now suppose that you keep the distance constant, while the rate and time vary. For example, if you fix the distances
traveled at 300 miles, then the equation 300 = rt would describe this problem situation. The faster you drive, the less time
the trip will take. Conversely, the slower you drive the more time the trip will take. Since 300 is a constant, as one variable
increases, the other must decrease. This is an example of inverse variation. The equation for inverse variation takes the
form
xy  k
or
y
k
, where k is the constant of variation.
x
The graph looks like the reciprocal function
y
1
, except
x
the value of k can be a value other than 1.
EX 1 Direct Variation: Hooke’s Law for springs states that the distance a spring is stretched (or compressed) is directly
proportional to the force on the spring. A force of 20 lbs stretches a certain spring 5 inches. Find a mathematical model
that relates the distance the spring is stretched to the force applied to the spring. How far will a force of 30 lbs stretch
the spring?
SOLUTION: An equation that describes this situation is
s  kf
where k is the constant of variation, s is the distance the
spring is stretched, and f is the force applied. Since a force of 20 lbs stretches a certain spring 5 inches, then solving the
equation
s
1
f
4
5  20k
will give the constant of variation, which is ¼ . So the mathematical model describing this situation is
. To find the stretch on the spring caused by 30 lbs of force, substitute 30 in for f and solve for s:
The spring will stretch
s
1
30 .
4
1
30 , or 7.5 in.
4
EX 2 Inverse Variation: The marketing department of a large company has found that the demand for one its product
varies inversely with the price of the product. When the price of the product is $7.50, the monthly demand is 50,000 units.
Approximate the monthly demand if the price is reduced to $6.
Another type of variation relates one variable to the power (or root) of another variable. For example, we can say that the
area of a circle varies directly as the square of its radius, or
variation in this case is
.
A  kr 2 , but we know from geometry that the constant of
EX 3 The period of a pendulum (the time for a complete back-and-forth trip) is directly proportional to the square root of
the length of the pendulum. If a pendulum 64 cm long has a period of 1.6 sec, what is the period of a pendulum 1 m long?
Some applications of variation involve problems with both direct and inverse variation in the same model.
EX 4 A company determines that the demand for one of its products is directly proportional to the amount spent on
advertising and inversely proportional to the price of the product. When $40,000 is spent on advertising, and the price per
unit is $20, the monthly demand is 10,000 units.
a) Write an equation of the variation described using k as a constant of variation.
b) If the amount spent on advertising was increased to $50,000 and the price per unit was reduced to $16, calculate the
monthly demand for the product.
To describe a variation of two different direct variations in the same
statement, the word “jointly” is used. We say that the model
z  kxy
can be described by saying that z is jointly proportional to x and y. For
example, we know from geometry that the volume of a cylinder varies
jointly as the square of the radius and the height of the cylinder:
JOINT VARIATION
The statements below are equivalent:
1.
z  kxy
for some constant k
2. z varies jointly as x and y
3. z is directly proportional to x and y
V  kr 2 h  r 2 h .
Give an equation of the variation described. Use k for the constant of variation.
1. V varies directly as the cube of r.
2. I is inversely proportional to the square of y.
3. h is directly proportional to t and inversely proportional to z+1.
4. A is jointly proportional to l and w.
5. w varies jointly as x and y and inversely as z.
6. m varies directly as n and inversely as z.
MORE EXAMPLES:
5. Suppose r varies jointly as t and s and inversely as the square of v. When t=3, s=18, and v=5, r = 3.78. Find r when t=4,
s=12, and v=4.
6. The heat loss through a glass window varies jointly as the area of the window and the difference between the inside and
3m 2 is 720 BTU when the temperature difference is 15
2
degrees Celsius, what is the heat loss through a window with area 4.5m when the temperature difference is 12 degrees
outside temperatures. If the loss through a window with area
Celsius?
7. The frequency of vibration of a piano string varies directly as the square root of the tension on the string and inversely
as the length of the string. The middle A string has a frequency of 440 vibrations per second. Find the frequency of a
string that has 1.25 times as much tension and is 1.2 times as long.