19 At a certain college, the economics club plans to sell T-shirts as a fund raiser. "e president of the club estimates that when the price of a T-shirt is $18, there will be 60 T-shirts sold, and for every $1 the price of the shirt is reduced, 10 additional T-shirts will be sold. Based on this estimate, what is the maximum amount of revenue, in dollars, the club can earn from T-shirt sales? (Revenue equals price times number sold. Disregard the $ sign when gridding your answer.)
Answers
Given : When the price of a T-shirt is $18, there will be 60 T-shirts sold,
For every $1 the price of the shirt is reduced, 10 additional T-shirts will be sold.
To Find : Maximum amount of revenue, in dollars, the club can earn from T-shirt sales
Solution:
Price of T shirt = 18 $
Number of T Shirts sold = 60
Revenue = 18 * 60 = 1080 $
For every $1 the price of the shirt is reduced, 10 additional T-shirts will be sold.
Let sat price of T Shirts reduces by x $ then increase in sell = 10x
T shirts price = 18 - x $
T shirts sold = 60 + 10x
Revenue R(x) = (18 - x)(60 + 10x)
R(x) = (18 - x)(60 + 10x)
R'(x) = (18 - x) (10) + (-1)(60 + 10x)
= 180 - 10x -60 - 10x
= 120 - 20x
R'(x) = 0
=> 120 - 20x = 0
=> x = 6
R''(x) = - 20 < 0
Hence Maximum revenue when x = 6
R(x) = (18 - x)(60 + 10x)
x = 6
=> R(6) = (18 - 6) (60 + 10 *6)
= (12)(120)
= 1440 $
Maximum revenue = 1440 $
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