The cost function for a telephone manufacturing plant is... Determine the following...

Question

Question #1:

The cost function for a telephone manufacturing plant is  and the revenue function is , where x is the number of telephones sold, in thousands.

a) Determine the profit function for the number of phones sold.

b) Determine the number of telephones sold that maximizes profit.

Answer 1

P(x) = R(x) - C(x) = -2x^2 +27x -5x -24 = -2x^2 + 22x -24

P(x)= -2x^2 +22x +24

take the derivative, set equal to zero, solve for x

P'(x) = -4x +22 = 0

x = 22/4 = 5.5 thousand = 5500 telephones

x= 5,500 telephones maximize profit

plug 5.5 into the original Profit function to get max profit

P(5.5) = -2(5.5)^2 +22(5.5) + 24 = -60.5 + 121 +24 = 84.5 = maximum profit

P(5) = -2(5)^2 + 22(5) +24 = -50+110 +24 = 84

P(6) = -2(6)^2 +22(6) +24 = -72 + 132 +24 = 84

the Profit function is a downward opening parabola with vertex = maximum point = (5.5, 84.5) = (x, P)