The total cost, in dollars, to produce x bins of cat food is given by C=16x + 6120.

Question

The revenue, in dollars, is R= -2x^2+258x

Find the profit. (Recall that profit is revenue minus cost.)

P =

At what quantity is the smallest break-even point? In other words, how many bins of cat food, x, must be produced and sold to make the profit equal to zero? 

Answer 1

R = C

-2x^2 + 258x = 16x + 6120

2x^2 - 242x + 6120 = 0

x^2 -121x + 3060 = 0

x = 121/2 + (1/2)sqr(121^2 -4(3060))

= (121+49)/2 or (121-49)/2 = 85 or 36

Break even out put levels are 36 or 85

(85+36)/2 = 121/2 = 60.5 = profit maximizing out put level

the cost function is a straight line with y intercept 6120 and slope =16. steep, upward sloping

Revenue is a downward opening parabola. They intersect twice, at x=36 and x=85

Profit maximizing output is where R'-C' = 0

-4x + 258 - 16 = 0

4x = 242

x = 242/4 = 60.5

P(x) = P(60.5) = R(60,5) - C(60.5) = -2(60.5)^2 + 258(60.5) - 16(60.5) - 6120

P = - 7320.5 + 15,609 - 968 -6,120 = 1,200.5 profit = max profit

breakeven points are equidistant to 60.5