a Amy can produce china pitchers at a cost of 5 each Letting

(a) Amy can produce china pitchers at a cost of $5 each. Letting x be the number of pitchers produced, write a cost equation C(x).

(b) Amy sells her china pitchers based on how many she has produced in a week. This works in the following way: if Amy has produced x pitchers, she will sell each pitcher for 17 x dollars. Assuming that Amy sells all of the pitchers she produces, write a revenue function in terms of x (where x is the number of pitchers produced and sold).

(c) Find the number of pitchers that she should produce and sell to maximize profit. Find the maximum profit.

Solution

a) Letting x be the number of pitchers produced, write a cost equation C(x).

C(x) = x*5 = 5x

b) if Amy has produced x pitchers, she will sell each pitcher for 17 x dollars.

R(x) = x*( 17 -x)

c) Profit = R(x) - C(x) = x(17 -x) -5x = -x^2 +17x -5x = -x^2 +12x

Its a quadratic function maximum value occurs at x = -b/2a = -(12)/2*(-1) = 6

Number of pitchers for maximum profit = 6

Maximu Profit , plug x= 6 in Profit function

Profit = -(6)^2 +12*6 = -36 +72 = $ 36