Chapter 8.4, Problem 38AYU

To determine

To find: The area of the shaded region.

Answer to Problem 38AYU

The area of the shaded region.

Explanation of Solution

Given:

The given diagram is in Figure 1

  

Figure 1

Calculation:

Consider the relation between the sides of the triangle is,

  $P{R}^2=Q{R}^2+P{Q}^2$

Then,

  $\begin{array}{l}{\left(10\right)}^2=Q{R}^2+{\left(8\right)}^2\\ Q{R}^2=100-64\\ QR=6\end{array}$

Then, the area of the triangle is,

  $\begin{array}{c}A=\frac{1}{2}\left(6\right)\left(8\right)\\ =24\end{array}$

Consider the area of the semi-circle is,

  $A=\frac{1}{2}\pi {\left(\frac{d}{2}\right)}^2$

Then,

  $\begin{array}{c}A=\frac{1}{2}\pi {\left(\frac{10}{2}\right)}^2\\ =\frac{25}{2}\pi \end{array}$

Consider the area of the shaded region is the difference of the area of semicircle and the area of the triangle, that is,

  $\begin{array}{c}A=\frac{25}{2}\pi -24\\ =15.72{\text{in}}^2\end{array}$