The revenue and cost functions for a particular product are

The revenue and cost functions for a particular product are given below. The cost and revenue are given in dollars, and x represents the number of units .

R(x) = 0.25x² + 182.5x

C(x) = 60x + 14106.25

(a) Find the profit function.
P(x) =

(b) How many items must be sold to maximize the revenue?

(c) What is the maximum revenue?

(d) How many items must be sold to maximize the profit?

(e) What is the maximum profit?

(f) At what production level(s) will the company break even on this product? (Enter your answers as a comma-separated list.)
____ units

Solution

a)

Profit = Income -    Cost

P(x) =     R(x)   -    C(x)

P(x) =( 0.25x2 + 182.5x) - (60x + 14106.25)

= -0.25x^2+ 122.5x - 14106.25

b) Tomaximise revenue: find dR/dx =-0.50x +182.5=0

x = 182.5/0.50 = 365 units

c) Max. Revenue = -0.25(365)^2 +182.5*365 = $33306.25

d) To maximise profit find dP/dx = -0.5x +122.5 =0

x = 245 units

d)