Applications of Linear Functions
PreCalculus H
Recall: A linear function is uniquely characterized by the fact that it has a constant rate of change.
.
We call that rate of change
Example 1 The radius is a bone that connects the elbow and the wrist.
A woman with a radius 22 cm long should be about 160 cm tall and a woman with a radius 26 cm long should be about 174 cm tall.
Write a linear equation relating height, y, to length of the radius x.
(a)
(b)
About how long should the radius be of a woman who is 157 cm tall?
1
Example 2 In 1980, The U.S. had 5.6% of the world‛s population. By 1990, the percent had dropped to 5.2%.
Assuming that our share of the world‛s population continues to drop at the same rate,
write a linear equation in slope-intercept form that expresses the U.S. percent of the world‛s population, y,
in terms of the year, x.
2
The following problems belong to a category called “cost function” problems.
Example 3 The Fluff Detergent Company manufactures detergent with a fixed overhead of $5000. Each box of Fluff Detergent
costs $0.40 to produce. Write an equation that represents the total cost, C(x), of producing x boxes of Fluff. This equation is
called the cost function.
3
Example 4 The Yummy Candy Company can produce 100 candy bars at a cost of $150 and 600 candy bars at a cost of $200.
Find the cost function.
What is the fixed cost (overhead) of producing candy bars?
4
Example 5 It costs Crunchy Cereal Corporation $1050 to produce 100 boxes of cornflakes and
$1250 to produce 500 boxes of cornflakes. Find the cost function. Sketch the graph.
2000
R(x)
1500
Money (in dollars)
C(x)
1000
500
x
600
400
200
Number of boxes of cornflakes
800
Suppose each box of cornflakes sells for $2.50. Write the revenue function, R(x), which describes the income generated from the
sale of x boxes of cornflakes. Sketch its graph on the same axes.
How many boxes of cereal must be sold for the Crunchy Cereal Corp. to “break even?”
Note: Break even point is where
.
If Profit = Revenue - Cost
Write a profit function, P(x), for the Crunchy Cereal Corp.‛s sale of cornflakes.
For what volume of sales will the Crunchy Cereal Corp. turn a profit?
What profit is made from selling 200 boxes of cornflakes?
5