Chapter 4.3, Problem 55PPS

To determine

To calculate: The range of the production of movie screens that will guarantee that the company will not lose money.

Answer to Problem 55PPS

The range of the production of movie screens that will guarantee that the company will not lose money is $\text{16 to 32}$ .

Explanation of Solution

Given information:

The profit function for the company that plans to build a large multiplex theatre $\text{P}\left(x\right)=-x^2+48x-512$ , where x is the number of movie screens and $\text{P}\left(x\right)$ is profit earned in thousands of dollars.

Formula used:

Solution of a polynomial equation is calculated using middle term splitting method and then common factors are taken out and simplified.

Calculation:

Consider the given information that “The profit function for the company that plans to build a large multiplex theatre $\text{P}\left(x\right)=-x^2+48x-512$ , where x is the number of movie screens and $\text{P}\left(x\right)$ is profit earned in thousands of dollars.”

The range will be the solution set of the profit function of the company. The company will be profitable between the range that makes the polynomial zero.

Recall that solution of a polynomial equation is calculated using middle term splitting method and then common factors are taken out and simplified.

Apply it,

Factorize $x^2-48x+512$ to find the range,

  $\begin{array}{c}{x}^2-48x+512=x^2-32x-16x+512\\ =x\left(x-32\right)-16\left(x-32\right)\\ =\left(x-32\right)\left(x-16\right)\end{array}$

Therefore, either $\begin{array}{c}x-32=0\text{ or }x-16=0\\ x=32\text{ or }x=16\end{array}$ .

Thus, the range of the production of movie screens that will guarantee that the company will not lose money is $\text{16 to 32}$ .