**QUIZ #7 SOLUTION
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A bookstore can obtain a certain gift book from the publisher at a
cost of $3 per book. The bookstore has been offering the book at a
price of $15 per copy and, at this price, has been selling 200 copies
a month. The bookstore is planning to lower its price to stimulate
sales and estimates that for each $1 reduction in the price, 20 more
books will be sold each month. At what price should the bookstore
sell the book to generate the greatest possible profit?
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Let x be the price of the book ( )
and N be the number of books sold
at that price. Then
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To maximize P(x), find the derivative:
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This always exists, and is zero when x=14. To see that this is
where P(x) attains an absolute maximum, note that P''(x)=-40 is
always negative on the domain, so P(x) is always concave down.
Therefore the price of the book should be chosen to be $14.00.
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Another way to do this problem would be to let x equal the amount of
reduction in price. Then N=200+20x, C=3N=600+60x,
, and . Maximizing
P yields x=1, representing a $1 reduction in price from $15 to
$14.
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Wed Nov 4 10:27:26 EST 1998