Chapter 9.5, Problem 7STP

The area of a rectangular television screen is 3456 $i{n}^2$.

The width of the screen in 24 inches longer than the length. What is a quadratic equation that represents the area of the screen? What are the dimensions of the screen?

To determine

To find : the quadratic equation that represents the area of the screen and what are the dimension of the screen

Answer to Problem 7STP

The quadratic equation is $l^2+24l-3456=0$ and the dimensions of the screen is 120 inches.

Explanation of Solution

Given information :

The area of the rectangular television screen is. $=3456{\text{ in}}^2.$ . The width of the screen is 24 longer than the length.

Formula used :

The area of the rectangular $=\text{ Lenght}\times \text{Breadth}$

Calculation :

Let $l$ be the length of the rectangular television screen.

The width of the rectangular television screen $=\left(l+24\right)\text{ inches}$

The quadratic equation is

  $\begin{array}{l}3456=l\left(l+24\right)\\ 3456=l^2+24l\\ l^2+24l-3456=0\end{array}$

Therefore, the quadratic equation is $l^2+24l-3456=0$

  $\begin{array}{l}{l}^2+24l-3456=0\\ l^2+72l-48l-3456=0\\ l\left(l+72\right)-48\left(l+72\right)=0\\ \left(l+72\right)\left(l-48\right)=0\end{array}$

  Either       or

  $\begin{array}{l}l+72=0\\ l=-72\end{array}$  $\begin{array}{l}l-48=0\\ l=48\end{array}$

Therefore, the length of the rectangular television screen is $48\text{ inches}$ because length cannot be negative.

Therefore, The width of the rectangular television screen

  $\begin{array}{l}=\left(l+24\right)\text{ inches}\\ \text{=}\left(48+24\right)\text{ inches}\\ =72\text{ inches}\end{array}$

The dimensions of the screen

  $\begin{array}{l}\text{=}\left(48+72\right)\text{ inches}\\ \text{=120 inches}\end{array}$

Hence, the quadratic equation is $l^2+24l-3456=0$ and the dimensions of the screen is 120 inches.