Please help with this word problem!

Question

A small theater has a seating capacity of 2000. When the ticket price is $18, attendance is 1500. For each $1 decrease in price, attendance increases by 100.

 

(a) Write the revenue R of the theater as a function of ticket price x.
R(x) =

(b) What ticket price will yield a maximum revenue?
$


What is the maximum revenue?

Answer 1

Let x = number of $1 decreases in price

 

Revenue = R(x) = (price)(quantity)

 

                      = (18 - x)(1500 + 100x)

 

The graph of R(x) is a parabola opening downward.  The graph has x-intercepts when 18-x = 0 and when 1500 + 100x = 0.

 

So, the x-intercepts are 18 and -15.

 

By the symmetry of the graph, the highest point of the graph lies halfway between the x-intercepts (x = (18 + (-15))/2 = 3/2 = 1.5)

 

Maximum revenue = R(1.5) = (16.5)(1650) = $27,225

 

If the ticket price is $16.50, the revenue will be maximized.