Chapter 8.3, Problem 33PPS

To determine

To find: The expression which represents the area of the shaded region.

Answer to Problem 33PPS

  $4\pi x^2+12\pi x+9\pi -3x^2-5x-2$

Explanation of Solution

Given:

  

Calculation:

In the given figure, the radius of circle is $2x+3$

So, the area of the circle is given by

  $\begin{array}{c}{A}_c=\pi {\left(2x+3\right)}^2\\ =\pi \left(4x^2+12x+9\right)\\ =4\pi x^2+12\pi x+9\pi \end{array}$

And the area of the rectangle is given by

  $\begin{array}{c}{A}_r=\left(x+1\right)\left(3x+2\right)\\ =3x^2+2x+3x+2\\ =3x^2+5x+2\end{array}$

Therefore, the area of the shaded region is given by

  $\begin{array}{l}{A}_c-A_r\\ =4\pi x^2+12\pi x+9\pi -3x^2-5x-2\end{array}$

Therefore, the expression that represents the area of the shaded region is $4\pi x^2+12\pi x+9\pi -3x^2-5x-2$