Question

1 x+14,000. The cost of producing x vans is given by the function 50 The CarryltAll mini-van, a popular vehicle among soccer moms, obeys the demand equation p C(x) 11160x+ 30,000 a) Express the revenue R as a function of x b) Express the profit P as a function of x c) Find the value of x that maximizes profit. What is the maximum profit? d) What price should be charged in order to maximize profit?

Answer 1

a) p = -(1/50)x + 14,000

R = p * x = -(1/50)x² + 14,000x

b) Profit = R - C = -(1/50)x² + 14,000x - (11160x + 30,000)

             = -(1/50)x² + 14,000x - 11160x - 30,000

             = -(1/50)x² + 2840x - 30,000

c) Profit is maximized at the quantity of x where first order derivative of profit function is 0.

-(2/50)x + 2840 = 0

(2/50)x = 2840

x = 2840 / (2/50) = 2840 * (50/2) = 71,000

Profit = -(1/50)x² + 2840x - 30,000

        = -(1/50)(71,000)² + 2840(71,000) - 30,000

       = - 100,820,000 + 201,640,000 - 30,000

       = $100,790,000

d) p = -(1/50)x + 14,000

       = -(1/50)(71,000) + 14,000

       = $12,580