How to Maximize Profit for Book Publisher?

[lullaby784]

1. A publisher wants to dispose books. For 400 copies or less the price is $30 per book. For orders of more than 400 the price of each book is dropped by 2 cents for each extra book ordered beyond 400. The cost of production is $6 for each book. Find a formula for the profit function and how many books must be sold to maximize profit?



2. P (x)= R(x)-C(x)

 

[tiny-tim]

welcome to pf!

hi lullaby784! welcome to pf!

translate the question into maths …

show us how far you've got, and where you're stuck, and then we'll know how to help!

 

[lullaby784]

This is what I have so far. I'm confused with what to do when the problem says more or less.

P(x)= (400+30x)(400-.02x) - 6x


To find the maximium profit I know to just find the first derivative of the profit function.

 

[lullaby784]

I really need help guys.

 

[tiny-tim]

hi lullaby784!

P(x)= (400+30x)(400-.02x) - 6x

i don't understand this at all

start by finding the total selling price for x books, and write separate equations for the two case x ≤ 400 and x > 400

 

[lullaby784]

Ok, so it would be. All I need are the formulas for profit function. The first profit function is for 400 copies or less and the second profit function is for greater than 400 copies. I'm also taking into the prices $30 and a decrease of $0.02 for each extra book over 400 copies.

P(x) = (x ≤ 400)($30)
($30x + $12000) - Cost
($30x + $12000) - $6x
P(x) = $24x + $12000
P(x) = (x > 400)(x-$0.02) - Cost
(x^2 + $399.98x - 8) - $6x
P(x) = x^2 +$393.98x - 8

Then I would just take the first derivative to find out how many books must be sold to maximize profit and to find the maximum profit.

 

[tiny-tim]

i'll do the first half for you …

for x ≤ 400, the selling price for each book is $30, so the profit on each book is $30 minus $6 = $24, so the total profit is $24x

ok, now what is the selling price for each book if x > 400 ?

(for example, we know that if x = 401, the selling price for each book is $29.98, if x = 402, the selling price for each book is $29.96 …)

 

[lullaby784]

Ok, seems like I had the first part correct already.

Total copies to maximize is 400, total profit to maximize is $9600

For x > 400. If the number of books was 401, then the profit function would be:

29.98-6=$23.98x (will give me my total profit)

Total copies to maximize is 401, total profit to maximize is $9615.98 Am I correct?

 

[tiny-tim]

(for example, we know that if x = 401, the selling price for each book is $29.98, if x = 402, the selling price for each book is $29.96 …)

For x > 400. If the number of books was 401, then the profit function would be:

29.98-6=$23.98x (will give me my total profit)

you're only repeating what i said …

you need to find a general formula, for the selling price for one book, in terms of x

(and btw, i have no idea what this means …)

Total copies to maximize is 401