Yahoo Answers 04-13 pt 3

3. The fixed and variable costs functions for the manufacture of speakers are FC(X)=125 and VC(X)10x^2-1/4x^3 while it's average revenue is given by AR(X)=250-5x. a. write the equation for cost, revenue, and profit. b. find the output level of x at which the profit is maximized. c. calculate the numbers of speakers which must be sold to maximize the revenue. d. find the manufacture's marginal revenue function.
a. Cost=Fixed Cost + Variable Cost or C(x)=FC(x)+VC(x)=10x^2-0.25x^3+125 Revenue=Average Revenue * Units Sold or R(x)=x(250-5x)=250x-5x^2 Profit=Revenue-Cost or P(x)=R(x)-C(x)=250x-5x^2-(10x^2-0.25x^3+125)=250x-15x^2+0.25x^3-125 b. The profit will be maximized where the derivative=0 P '(x)=250-30x+0.75x^2 .75(x^2-40x+333.33)=0 x^2-40x+333.33=0 (x^2-40x+400)+333.33-400=0 (x-20)^2=66.67 x-20=+/- 8.16 x=11.84 or x=28.16 hmm. there are two zeros P(11)=1142.75 P(28)=603 Max profit is 11 units c. Max revenue is where the derivative of the revenue=0 R '(x)=250-10x 250-10x=0 x=25 d. Marginal Revenue is the derivative of the revenue (see part c)