From past experience, a publisher knows that a particular type of book will sell 14,000 copies at a price of $20 each. Market research further indicates that for every one-dollar increase in price, sales will fall by 400 copies. The publisher asks for your advice in deciding the suggested retail price of the next book published.1
From a symbolic analysis of the situation, a function used to model this situation is given by R(x) = (14000 - 400x)(20 + x), where x represents the number of increases and R(x) is the anticipated revenue. A table of data may be constructed by substituting values for x into the expression and calculating values for R(x).
Revenue Data
Increases
(x)Books Sold
(14000-400x)Price of Book
(20+x)Revenue
(14000-400x)(20+x)0
14,000
$20
$280,000
1
13,600
$21
$285,600
2
13,200
$22
$290,400
3
12,800
$23
$294,400
4
12,400
$24
$297,600
5
12,000
$25
$300,000
6
11,600
$26
$301,600
7
11,200
$27
$302,400
8
10,800
$28
$302,400
9
10,400
$29
$301,600
10
10,000
$30
$300,000
11
9,600
$31
$297,600
12
9,200
$32
$294,400
13
8,800
$33
$290,400
14
8,400
$34
$285,000
15
8,000
$35
$280,000
16
7,600
$36
$273,600
17
7,200
$37
$266,400
18
6,800
$38
$258,400
19
6,400
$39
$249,600
20
6,000
$40
$240,000
As can be seen from the table above, the revenue without increasing the suggested retail price of the book would be expected to be $280,000. For the first few times the price is increased the revenue also increases. The revenue reaches a maximum value of $302,400 for seven and eight increases.
Further analysis of this situation may be accomplished by graphing the function listed above or by a symbolic analysis. The numerical analysis may be modeled effectively on the TI-83 graphics calculator.
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Problems of this type may be found in Mathematical Analysis, 3rd. Ed. By Arya and Lardner, published by Prentice Hall, 1989, page 93. |