Chapter 8.2, Problem 68STP

To determine

To find: the area of the region.

Answer to Problem 68STP

The area of the region is $63.25{\text{cm}}^2$ .

Explanation of Solution

Given information:

Form the figure provided in the question, it can be observed that,

Base of the triangle is $8\text{cm}$ and height of the triangle is $\text{6cm}$ .

Calculation: to find area of the region, first find the radius of the semicircle,

To find the radius of the semicircle use the Pythagoras theorem,

Let $r$ is the radius of the semicircle,

  $\begin{array}{l}{\left(2r\right)}^2={\left(6\right)}^2+{\left(8\right)}^2\\ 4r^2=36+64\text{ [square]}\\ 4r^2=100\text{ [add]}\\ r^2=25\text{ [divide]}\\ r=5\text{ [sqaure root]}\end{array}$

Now, the area of the region is sum of the area of the triangle and area of semicircle,

  $\begin{array}{l}\text{area}=\frac{1}{2}\left(6\right)\left(8\right)+\frac{\pi {\left(5\right)}^2}{2}\\ \text{area}=24+12.5\left(3.14\right)\text{ [divide]}\\ \text{area}=63.25{\text{cm}}^2\text{ [multiply and add]}\end{array}$

Thus, the area is $63.25{\text{cm}}^2$ .