Question

Question 4 20 pts A firm believes a product's sales volume (S) depends on its unit selling price (P) as S $100 P. The production cost C is $993+9S. Determine the sales volume (S) at which the firm's profit is a maximum. (Hint: calculate derivative of the profit function with respect to the price) Margin of error: +/-2 (answers within range [x-2, x+2], where x is the true answer, are considered correct)

Answer 1

Profit = Total Revenue - Total Cost

S = 100 - P

P = 100 - S

TR = Price x Quantity = (100 - S) x S = 100S - S²

TC = 993 + 9S

Profit = TR - TC

$\pi$ = 100S - S² - 993 - 9S

$\pi$ = 91S - S² - 993

$\frac{\partial \pi }{\partial S}=91-2S=0$

2S = 91

S = 45.5

$\frac{\partial^2 \pi }{\partial S^2}=-2<0$

Therefore, profit is maximised at S = 45.5