derivative hot dog problem

Question

mr. hot and mrs. dog are taking over a hot dog stand. the current owner now charges $2 per hot dog and gets appx 56 customers during the lunch hour. they want to figure out how to maximize their profits. they do some research and find that for every $0.50 they raise the price of the hot dogs, they will lose 2 customers.

using derivatives, what is the maximum amount they can charge per hot dog so their income is maximized and what is that maximum income?

Answer 1

Let C = # customers and P = Price

C = 64 - 4P

I = Income = PC = 64P-4P²

d/I/dP = 64- 8P = 0 to maximize income

P to generate max income is 8 and C = 32 and max income will be 256 (as opposed to 112 now!)

This question doesn't seem to be too realistic, i.e. quadruple the price will not quite cut customers in 1/2 and still maximize income,