Chapter 4.1, Problem 42PPS

a.

To determine

Find the ticket price for the greatest income of the Drama Club.

Answer to Problem 42PPS

  $\$3.5$

Explanation of Solution

Given information:

Around $300$ people attended the winter play performed by the Franklin High School’s Drama Club. The price of the ticket was $\$8$ . An increase of $\$1$ on each ticket price, people decreases by 20.

Calculation:

Here, we will assume $x$ as the number of price that maximizes the income and $f\left(x\right)$ be the function.

Thus, the amount and the number of people will be $\$\left(8+x\right)and\left(300-20x\right)$ .

Now,

  $\begin{array}{l}f\left(x\right)=\left(8+x\right)\left(300-20x\right)\\ f\left(x\right)=2400-160x+300x-20x^2\\ f\left(x\right)=-20x^2+140x+2400\end{array}$

As, the function is in parabolic form:

  $\begin{array}{l}x=-\frac{b}{2a}\\ x=-\frac{140}{2\left(-20\right)}\\ x=\frac{140}{40}\\ x=3.5\end{array}$

Thus, the maximum value of the function is attained at $x=3.5$ .

Hence, the ticket price for the greatest income is $\$3.5$ .

b.

To determine

Find the income if the price of tickets are raised.

Answer to Problem 42PPS

  $\$2645$

Explanation of Solution

Given information:

Given: Around $300$ people attended the winter play performed by the Franklin High School’s Drama Club. The price of the ticket was $\$8$ . An increase of $\$1$ on each ticket price, people decreases by 20.

Calculation:

Here, the quadratic function will be:

  $f\left(x\right)=-20x^2+140x+2400$

As, it is in the parabolic form the maximum value would be attained at $x=3.5$ .

Now,

  $\begin{array}{l}f\left(3.5\right)=-20{\left(3.5\right)}^2+140\left(3.5\right)+2400\\ =2645\end{array}$

Hence, the expected income is $\$2645$ .