Optimization Problem

[smgrojean]

I am having trouble being sure about if I have properly optimized the following problem:

"A bake sale has 100 cookies to sell. All the cookies will be sold if they cost 25 cents. For each 5 cents increase in cost, 4 fewer cookies will be sold. Find the price that will maximize the bake sale's profit and the resulting profit."

I started by saying the goal would be to maximize the profit (P) as a function of cookie cost (c). With N being the number of cookies

Then I said that:

N(c) = 100 - (4/5) (c-25)
= 100 - (4/5)c + 20
= 120 - (4/5)c

then I set up P(c)

P(c) = N(c)(c-5)
= (120 - (4/5)c)(c-5)
= -(4/5)c^2 + 124(c) - 600

Then I took the derivative:

P'(c) = 124 - (8/5)c

The critical point is at 155/2 or 77.5

I got the best price being 77.5 cents. Did I do that all correctly? Also, how do I then figure out the profit?

 

[girodd]

I am having trouble being sure about if I have properly optimized the following problem:

"A bake sale has 100 cookies to sell. All the cookies will be sold if they cost 25 cents. For each 5 cents increase in cost, 4 fewer cookies will be sold. Find the price that will maximize the bake sale's profit and the resulting profit."

I started by saying the goal would be to maximize the profit (P) as a function of cookie cost (c). With N being the number of cookies

Then I said that:

N(c) = 100 - (4/5) (c-25)
= 100 - (4/5)c + 20
= 120 - (4/5)c
This is correct so far

then I set up P(c)

P(c) = N(c)(c-5)

The "profit" (revenue actually, we know price but not cost) is number sold times price charged, N(c)*c, not c-5. Note that your profit is in cents, not dollars.

= (120 - (4/5)c)(c-5)
= -(4/5)c^2 + 124(c) - 600

Then I took the derivative:

P'(c) = 124 - (8/5)c

The critical point is at 155/2 or 77.5

I got the best price being 77.5 cents. Did I do that all correctly? Also, how do I then figure out the profit?

Assuming you determine the optimum price to charge, you just plug that value into your profit expression.