Calculus business is application

Question

 An Italian company can manufacture at most 200 leather wallets per month. Market research suggests that about x wallets per month can be sold at the price p(x) = (500 – x) eurosand the cost of producing x wallets per week is C(x) = 150 + 4x + x2 euros. Find:

a. Revenue from sale of x wallets per week

b. Maximum weekly profit and the number of wallets produced to obtain the maximum

d. Marginal Revenue at Maximum weekly profit level

c. Marginal cost functions at Maximum weekly profit level

 

Answer 1

revenue = price times quantity sold = (500-x)x = 500x -x^2

profit = revenue - cost = 500x -x^2 - (150 + 4x + x^2) = 500x -x^2 -150 -4x -x^2 = 496x -2x^2 -150

P'(x) = 496 -4x = 0

4x = 496

x = 496/4 = 124 = profit maximizing output

max profit = (500-124)124 = 376(124) = 46,624 euros

max marginal revenue at maximum profit = max marginal cost at maximum profit

MR = 500-2x = 500-2(124) = 500-248= 252

MC = 4 +2x = 4 + 2(124) = 4+ 248 = 252

max marginal revenue = 252 euros

marginal cost function at max profit = MC = 4+2x = 4+2(124) = 252 euros