Chapter 13.4, Problem 61E

To determine

To find the increase in tickets.

Answer to Problem 61E

The tickets should be increased by $\$8.50$ to $\$11.50$ for a total profit of $\$2645.50$

Explanation of Solution

Given:

The average attendance is $400$ people and the cost of each ticket is $\$3$ . It is estimated that for every $\$1$ increase in ticket price, the attendance will decrease by $20$ .

Calculation:

Let $x$ be the increase in tickets. The profit can be modeled by the function

  $\begin{array}{l}P(x)=(3+x)(400-20x)\\ P(x)=-20x^2+340x+1200\end{array}$

The maximum profit at the vertex of this parabola. The x-coordinate of the vertex is at

  $x=-\frac{b}{2a}=-\frac{340}{-40}=8.5$

And the y-coordinate will be

  $P(8.5)=(3+8.5)(400-20\times 8.5)=2645$

The tickets should be increased by $\$8.50$ to $\$11.50$ for a total profit of $\$2645.50$