Math 127 Section 7.8 - Page 1
Section 7.8
Modeling Using Variation
I.
Direct Variation
A.
Example
1.
The amount of money that you get paid, P, varies directly with the number of hours, H,
that you work. If you get paid $370 for working 40 hours, how much will you get paid
for working 35.5 hours?
2.
First, we need to come up with a general equation:
P = kH
k is called the constant of proportionality. In this case, k does have another name,
the pay rate.
3.
Next, we need to substitute the initial conditions into the formula to find k:
370 = k(40)
Dividing both sides by 40, we find that k = 9.25.
4.
Now we need to rewrite the original formula, replacing k with 9.25:
P = 9.25H
5.
Now we can substitute into this formula for H to answer the question:
P = 9.25(35.5) = 328.375
Answer:
B.
II.
You will get paid $328.38.
In general, with direct variation, any time that you see the words “varies directly” or “is
directly proportional to”, write a formula with the first variable = k times the second variable.
Other Types of Variation
A.
Inverse Variation
1.
The words “varies inversely” or “is inversely proportional to” will appear in the
problem.
2.
The formula will be:
First variable =
k
nd
2 Variable
B.
Joint Variation
1.
The words “varies jointly” will be in the problem.
2.
Joint variation means that there are two or more direct variations (but only one
constant) in the problem.
3.
Remember that direct variation means multiplication.
4.
The formula will be:
1st Variable = k(2nd Variable)(3rd Variable)
© Copyright 2012 by John Fetcho. All rights reserved.
Math 127 Section 7.8 - Page 2
C.
Combined Variation
1.
Combined variation occurs when you have both a direct and an inverse variation in
the same problem. However, you will never actually see the words “combined
variation”.
2.
D.
Formula:
1st Variable =
k(Direct)
Inverse
Examples - Solve each variation problem.
1.
The water temperature of the Pacific Ocean varies inversely as the water's depth. At
a depth of 1,000 meters, the water temperature is 4.4o C. What is the water
temperature at a depth of 5,000 meters? (Page 569, #28)
We begin by defining our variables.
Let T = The temperature
Let D = The depth
So our variation formula will be:
k
T=
D
So we substitute our initial conditions into this formula and solve for the constant of
variation, k:
k
Multiply by 1,000
4.4 =
1000
k = 4,400
Now we can rewrite our variation formula, substituting in for k.
4,400
T=
D
Now we will substitute into this equation the values that we know to find the new
temperature.
4,400
= 0.88
T=
5,000
Answer:
2.
The temperature at a depth of 5,000 meters is 0.88o C.
One's intelligence quotient, or IQ, varies directly as a person's mental age and
inversely as that person's chronological age. A person with a mental age of 25 and a
chronological age of 20 has an IQ of 125. What is the chronological age of a person
with a mental age of 40 and an IQ of 80? (Page 570, #36)
We start by defining our variables.
Let IQ = The intelligence quotient
Let M = The mental age
Let C = The chronological age
Now we write a variation formula.
kM
IQ =
C
Substituting in our initial conditions, we will solve for the constant of variation.
k (25)
125 =
Multiply both sides by 20.
20
2,500 = k(25)
Divide both sides by 25.
k = 100
Rewrite the variation equation.
© Copyright 2012 by John Fetcho. All rights reserved.
Math 127 Section 7.8 - Page 3
100M
C
Substitute in the new values.
100(40) 4,000
Multiply both sides by C.
80 =

C
C
80C = 4,000
Divide both sides by 80
C = 50
IQ =
Answer: The person has a chronological age of 50 years .
3.
Now you try one: Page 570, #41
The average number of daily phone calls, C, between two cities varies jointly as the
product of their populations, P1 and P2, and inversely as the square of the distance, d,
between them.
a)
Write an equation that expresses this relationship.
Answer:
b)
kP1 P2
d2
The distance between San Francisco (population: 777,000) and Los Angeles
(population: 3,695,000) is 420 miles. If the average number of daily phone calls
between the cities is 326,000, bind the value of k to two decimal places and write the
equation of variation.
Answer:
c)
C=
k  0.02; C =
0.02P1 P2
d2
Memphis (population: 650,000) is 400 miles from New Orleans (population:490,000).
Find the average number of daily phone calls, to the nearest whole number, between
these cities.
Answer:
There are about 39,813 daily phone calls between Memphis and New
Orleans.
© Copyright 2012 by John Fetcho. All rights reserved.