Business mathethatics

Question

The estimated wholesale unit price (p) in dollars of the Pacific Hawaii Flour Factory to sell x cases of butter squash flour is: p = -0.4x + 580

The cost to produce x cases of flour is given by the function: C(x)= 264x + 45,000

a) Find the number of cases and price per case that will produce the maximum revenue

b) What is the maximum revenue?

c) Find the profit function P(x)?

d) What is the maximum profit realized and how many cases must be produced to realize this maximum profit?

Answer 1

TR = X*P(X) = X*(500-(4/10)X =500X- 4/10X²

MR= 500- (8/10)X

TR is maxed when MR= 0 or 500 = 8/10X or X= (10/8)*500 = 625

At this point TR = 500*625- (4/10)*(625²) = 156,250

π= TR-TC = 500X-(4/10)X² - 45,000- 264X = -45,000+236X -(4/10)X²

We can either differentiate the Profit function with respect to X , set it to 0 and solve for X or Use the Golden Rule (MR=MC) they are equivalent I will Use the Latter.

MC = dTC/dX = 264 (constant marginal cost)

MR=MC when 264= 500-(8/10)X NOTE: MR is twice as steep as the demand curve

so, 236= (8/10)X or X = (10/8)(236) 295. when X= 295, p = 500-(4/10)(295) = 382

TR = 295*382 = 112,690 TC = 45,000 +236*(295) = 114,620

They are losing money (112,690-114,620) = -1930

Hope this helps