Question

At Otto’s egg farm, Otto has found that when he charges $0.05 per egg, he sell so many eggs that he cannot keep up with the demand and has zero profit. When he charges $0.17 per egg, he sells so few eggs that he has to throw some out and is also left with zero profit. Currently he charges $0.15 per egg and has a profit of $124 per day. Let p(c)be his profit in dollars, from charging c dollars per egg. Assume that p(c) is a quadratic function of c.

Find a formula for p(c)
What should Otto charge to maximize his profit? What will that profit be?

Answer 1

a) P(c) = xc² + yc + z, and P(.15) = $124, P(.05) = 0, P(.17) = 0. Thus:

(#1) .0225x + .15y + z = 124

(#2) .0025x + .05y + z = 0

(#3) .0289x + .17y + z = 0

Subtracting #2 from #3: (#4) .0264x + .12y = 0

Subtracting #2 from #3: (#5) .02x + .1y = 124

Subtracting 1.2×#5 from #4: .0024x = -148.8 → x= -62000

Using this value of x in #5: -1240 + .1y = 124 → y = 13640

Using x and y in #1: -1395 + 2046 + z = 124 → z = -527

P(c) = -62000c² + 13640c - 527

b) To find the maximum of P(c), find a such that P'(a) = 0

-124000c + 13640 = 0

c = .11

At this price, his profit would be $223.20