Question

At a price of $8.5 per ticket, a musical theater group can fill every seat in their 1800 seat performance hall. For every additional dollar charged for admission, the number of tickets sold drops by 65. a) What ticket price maximizes revenue? Round your answer to the nearest cent.

Answer 1

Answer:

The ticket price that maximizes revenue is $18.10

Explanation:

Hi, first we need to construct the revenue equation in terms of the additional dollar charge (that would be X). That is:

$Revenue=Price*Quantity$

$Revenue=(8.5-X)(1,800-65X)$

So we expand it:

$Revenue=15,300-552.5X+1,800X-65X^{2}$

$Revenue=-65X^{2} +1247.5X+15,300$

This is a parabola, and we need to find its vertex, which in our case that would be the maximum additional dollar charge in order to obtain the highest revenue possible, to find the vertex, we need to consider that:

$Y(X)=AX^{2}+ BX+C$

And to find the X-coordenate we have to use the following equation.

$Vertex(X)=\frac{-B}{2A}$

In our case, A= -65; B= 1,247.5, so, all should look like this:

$Vertex (X)=\frac{-(1247.50}{2(-65)} =9.6$

That means, we need to make 9.6 increments of $1 in order to obtain the max revenue possible, therefore, the price would be

Price = $8.50 + $1(9.6)= $8.50 + $9.6 =$18.10

Best of luck.