I don't Understand How To Do This Problem

[nch]

A bookstore is deciding what price it should charge for a certain book. After research the bookstore finds that if the books price is p dollars (where p is <= to 26), then the number of books sold per month is 130-5p. What price should the store charge to maximize its revenue?

A full explanation would be very helpful

Thanks

 

[soroban]

Hello, nch!

A bookstore is deciding what price it should charge for a certain book.
After research the bookstore finds that if the books price is \(\displaystyle p\) dollars (where \(\displaystyle p \le 26\)),
then the number of books sold per month is \(\displaystyle 130-5p.\)
What price should the store charge to maximize its revenue?

The revenue is given by: .\(\displaystyle R \:=\: p(130-5p)\)

We have: .\(\displaystyle R \:=\:130p-5p^2\)

The graph is a parabola that opens downward \(\displaystyle \cap.\)
The maximum is at its vertex.

For a parabola, \(\displaystyle y \:=\:ax^2 + bx + c,\)
. . the formula for the vertex is: .\(\displaystyle x \:=\:\frac{\text{-}b}{2a}\)

We have: .\(\displaystyle R \:=\:\text{-}5p^2 + 130p\)
. . \(\displaystyle a = \text{-}5,\:b=130,\:c=0\)

The vertex is at: .\(\displaystyle p \:=\:\dfrac{\text{-}130}{2(\text{-}5)} \:=\:13\)

The book should be sold at $13 each.