A store that installs satellite TV receivers finds that if i

A store that installs satellite TV receivers finds that if it installs x receivers per week, its costs will be given by C(x) = 80x + 1950 and its revenue will be given by R(x) = -2x^2 + 240x, both in dollars. What is the fixed cost? What is the marginal cost? What is the smallest number of units that should be sold in order to break even? What is the maximum profit?

Solution

Costs : C(x) = 80x +1950

Revenue : R(x) = -2x^2 +240x

a) Fixed cost : C(0) = $1950

b) Marginal cost is the derivative of the cost function, so take the derivative

C\'(x) = $80

c) Break even: C(x) = R(x)

80x +1950 = -2x^2 +240x

2x^2 - 160x +1950 =0

x^2 -80x + 975 =0

x = 65 and x = 15

Out of the two smaller x = 15. So, 15 no. of units should be sold to breakeven

d) Maximum profit: Profit = R(x) - C(x) = -2x^2 +240x - (80x +1950 )

=-2x^2 +160x -1950

Maximum profit: x= -b/2a = -(160)/(-2*2) = 40 units

Maximum profit : Profit(40) = -2(40)^2 + 160*40 -1950 = $ 1250