Chapter 14, Problem 9MCE

To determine

To find: The correct option for the area of the region bounded by $\overline{AB}$ and $\stackrel{⌢}{AB}$ .

Answer to Problem 9MCE

The correct option is (c).

Explanation of Solution

Given information: The circle has center $O$ , $m\stackrel{⌢}{AB}=90$ , and $OA=6$ .

Calculation:

As $m\stackrel{⌢}{AB}$ cover the one-fourth area of the circle and the length of $OA$ , and $OB$ are equal to the radius of the circle. So, the area of the region bounded by $\overline{AB}$ and $\stackrel{⌢}{AB}$ is:

  $\begin{array}{c}A=\text{Area}\text{covered}\text{by}\stackrel{⌢}{AB}-\text{Area}\text{of}\Delta AOB\\ =\frac{1}{4}\pi r^2-\frac{1}{2}\times OA\times OB\end{array}$

The radius of the circle is 6. Substitute 6 for r , $OA$ , and $OB$ in the above formula.

  $\begin{array}{c}A=\frac{1}{4}\pi {\left(6\right)}^2-\frac{1}{2}\left(6\right)\left(6\right)\\ =\frac{36}{4}\pi -3\left(6\right)\\ =9\pi -18\end{array}$

Therefore, the correct option is (c).