Chapter 8.3, Problem 52PFA

To determine

To find: The value of x .

Answer to Problem 52PFA

  $x=12$

Explanation of Solution

Given:

The area of the shaded region is 214 square units.

  

Calculation:

The area of inner rectangle is given by

  $\begin{array}{l}\left(x+2\right)\left(x-3\right)\\ =x^2-3x+2x-6\\ =x^2-x-6\end{array}$

And the area of outer rectangle is given by

  $\begin{array}{l}\left(x+8\right)\left(x+5\right)\\ =x^2+5x+8x+40\\ =x^2+13x+40\end{array}$

So, the area of the shaded region is given by

  $\begin{array}{l}{x}^2+13x+40-\left(x^2-x-6\right)\\ =x^2+13x+40-x^2+x+6\\ =14x+46\end{array}$

Now, it has been given that the area of the shaded region is 214 square units. So,

  $\begin{array}{c}14x+46=214\\ 14x=214-46\\ 14x=168\\ x=\frac{168}{14}\\ x=12\end{array}$