Pre Calculus
Notes 1-8 Applications of Linear Functions
A linear function is uniquely characterized by the fact that it has a constant rate of change.
We call that rate of change
.
Ex 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.
(a) Write a linear equation relating height, y, to length of the radius x.
(b)
About how long should the radius be of a woman who is 157 cm tall?
Ex2 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.
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The following problems belong to a category called “cost function” problems.
Ex 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.
Ex 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?
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Ex 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
Money (in dollars)
1500
1000
500
600
400
200
Number of boxes of cornflakes
x
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 2000 boxes of cornflakes?
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Ex 6 Economists model supply and demand for a certain product with linear equations.
where y is the number of items produced
Consider the supply equation:
where y is the number of items sold
and the demand equation:
y = Number of Items
Graph these two equations on the same axes.
200
150
100
50
0
500
1000
1500
2000
p = selling price
What do the signs of the slopes tell us about the effect of price on supply and demand?
Where do the lines intersect? ___________________________________
This is called the equilibrium point. It is the point where supply = demand.
How does knowing the equilibrium point help the manufacturer of a good? the retailer?
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