The tickets to a charity concert will sell out if the price is $15 per ticket. it is estimated that for each $1 increase in ticket's price, 20 tickets will remain unsold. what ticket price will maximize the charity's profit, if there are total 500 seats in the concert hall?

Let's define variables:

x: number of times the ticket price increases ($ 1).

The income is given by:

I (x) = (15 + x) * (500-20x)

We can rewrite this function in the following way:

I (x) = - 20x ^ 2 + 200x + 7500

We derive the function to find the maximum:

I ' (x) = - 40x + 200

Equaling zero we have:

-40x + 200 = 0

We clear x:

x = 200/40

x = 5

Thus, the ticket price that maximizes the benefit is:

15 + x = 15 + 5 = 20 $

Answer:

A ticket price that will maximize the charity's profit, if there are total 500 seats in the concert hall is:

$ 20