Assume that if the price of a certain book is p dollars, then it will sell x copies where
x = 7000(1 − p/35). Suppose the dollar cost of producing those x copies is 15000 + 2.5x.
Finally, assume that the company will not sell this book for more than $35. Determine the
price for the book that will maximize proﬁt.

Question

Assume that if the price of a certain book is p dollars, then it will sell x copies where
x = 7000(1 − p/35). Suppose the dollar cost of producing those x copies is 15000 + 2.5x.
Finally, assume that the company will not sell this book for more than $35. Determine the
price for the book that will maximize proﬁt.

anonymous · Answer

PROFIT = INCOME - COST = p x - (15000 + 2.5x)
PROFIT = p(7000)(1-p/35) - (15000 + 2.5(7000-p/35)
PROFIT = 7000p - 200p^2 -15000 - 17500 + 0.071p
ignore last term as insignificant
set d(PROFIT)/dp = 0 = 7000 -400p
p = 7000/400 = 17.5 is maximum or minimum
take 2nd derivative = - 400, thus p = 17.5 is a maximum.

Check my calculations, tho, as it is late and I am off to bed.

The method is to get PROFIT(p) and then take the
first derivative to get extreme value and the second
derivative to see whether you have a mx or a min.