Question

At Otto’s egg farm, Otto has found that when he charges $0.05 per egg he sells so many eggs that he cannot keep up with 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 left parenthesis c right parenthesis be his profit in dollars from charging c dollars per egg. Assume that P left parenthesis c right parenthesis is a quadratic function of c . a) Find a formula for P left parenthesis c right parenthesis . b) What should Otto charge to maximize his profit? What would that profit be? Write your answers to these questions in a complete sentence.

Answer #1

Let P(c) = Ac^{2} +Bc + C

Acc. to question P(0.05) = 0

=> 0.0025A+0.05B + C=0

P(0.17) = 0

=>0.0289A+ 0.17B + C=0

P(0.15)= 124

=>0.0225A+0.15B + C=124

Solving above equations , we get

This gives A= -62000,B=13640, C= -527

a) Therefore **P(c) = -62000c ^{2}+ 13640c
-527**

b) For maximum profit, differnetiate above equation wrt c, we get

-124000c+13640 =0

=> c= 13640/124000

=> **c = 0.11**

The maximum profit be

= -62000*0.0121+13640*0.11- 527

= **$ 223.2 per day**

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