Question

rences Mailings Review View Help 7. A manufacturer has developed a new design for circuit panels. Marketing studies have shown that annual demand is estimated by the equation x = fp) = 200.000 - 500p, or p = 400 - 002x (for 0 < x < 200,000) where x # of units demanded and p = price unit in $. Suppose the cost function for the manufacturer is estimated as C(x)= 150,000 + 300x +.003x? a. Find the marginal cost function. (3) b. Find the revenue and marginal revenue functions. (4) Revenue function: Marginal Revenue function: c. Find the profit and marginal profit functions. (4) Profit function: Marginal Profit function:

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Answer 1

a. x = f(p) = 200,000 - 500p

p = 400 - 0.002x

C(x) = 150,000 + 300x + 0.003x²

Marginal cost function can be found by differentiation :

Differentiating w.r.t. x

C'(x) = 0 + 300 + 2*0.003x^2-1 = 300 + 0.006x

b. Total revenue = Price x Quantity = px = (400-0.002x)(x)

TR(x) = 400x - 0.002x²

Differentiating w.r.t. x

MR(x) = 400 - 2*0.002x = 400 - 0.004x

c. Profit function, P(x) = TR(x) - TC(x)

=400x - 0.002x² - ( 150,000 + 300x + 0.003x² )

=400x - 0.002x² - 150,000 - 300x - 0.003x²

P(x) =100x - 0.005x² - 150,000

Marginal Profit = Diff. w.r.t. x

P'(x) = 100 - 0.01x