Applied Mathematics for Business and Economics

Lecture Note Function

4 Functional Models

4.1 A Profit Function ......................................................................................

In the following example, profit is expressed as a function of the price at which a
product is sold.

Example 1
A manufacturer can produce radios at a cost of $ 2 apiece. The radios have been
selling for $ 5 apiece, and at this price, consumers have been buying 4000 radios a
month. The manufacturer is planning to raise the price of the radios and estimate that
for each $1 increase in the price, 400 fewer radios will be sold each month. Express
the manufacturer’s monthly profit as a function of the price at which the radios are
sold.

Solution
Begin by stating the desired relationship in words.
Profit = (number of radios sold) (Profit per radio)
Let x denote the price at which the radios will be sold andPx( )

x x)15(400)]5(10[

the corresponding

profit.
Number of radios sold = 4000 – 400 (number of $ 1 increases) the number of $
1 increases in the price is the difference x-5 between the new and old selling prices.
Hence,
Number of radios sold = 4000 – 400 (x - 5)
= 400 − − = −

Profit per radio = x – 2

Px( )=−−400 15( x x)( (^2) )

4.2 Functions Involving Multiple Formulas ...................................................

In the next example, you will need three formulas to define the desired
function.

Example 2
During a drought, residents of Marin Country, California, were faced with a severe
water shortage. To discourage excessive use of water, the country water district
initiated drastic rate increases. The monthly rate for a family of four was $ 1.22 per
100 cubic feet of water for the first 1,200 cubic feet, $10 per 100 cubic feet for the

1 2 3 4 5 x

700

600

(^500 )
350
300
200
100
0
(2,300) (3,350)
C(x) = 50x + 200