Determine the maximum profit and price that would yield the maximum profit for each.

Question

A) P=-450p2 + 13400p - 52,000

B) P=-375p2 + 8,900p - 15,000

C) P= -160p2 + 85,800p - 50,000

Answer 1

A) P = -450p^2 +13,400p - 52,000

take derivatives, set = 0, solve for p=price

P' = -900p +13,400 = 0

p = 13400/900 = 134/9 = $14.89 rounded off = Profit maximizing price

max Profit = -450(134/9)^2 + 13400(134/9) -52000 = -50(134^2)/9 +134^2(100/9 -52000

= 50(134^2)/9-52000 = 897800/9-468000= $449,800

B) P =-375p^2 + 8900p - 15,000

complete the square

P = -375(p^2 - 356p/15 + 356^2/30^2) -15000+375(356^2)/30^2

=-375(p-178/15)^2 +37,806.67 in vertex form

with vertex = ($11.87, $37,8806.67) rounded off to nearest cent

profit maximizing price = $11.87

max profit = $37,806.67

C) do same way, either way, like A) or B)

P = -160p^2 +85,800p - 50,000

=-160(p^2 -536.25p + 268.125^2) -50000 + 268.125^2

= -160(p - 268.125)^2 + 21891.02 in vertex form

max Profit = $21,891.02 rounded off to nearest cent

p= $268.13 = profit maximizing price rounded to nearest cent