Chapter 11.3, Problem 41PPS

To determine

The area of each shaded region.

Answer to Problem 41PPS

The area value is $\text{A}\left(\text{shaded}\right)=50.3{\text{in}}^2$

Explanation of Solution

Given information:

The area of each shaded region.

  

Formula used:

  $\text{A}\left(\text{shaded}\right)=\frac{1}{2}\text{A}\left(\text{lg}\text{circle}\right)-\frac{1}{2}\text{A}\left(\text{sm}\text{circle}\right)$

Calculation:

The larger circle has a radius of 6 in.  The three smaller circles are congruent and the sum of their diameters is 12 in. 

So, each has a radius of 2 in.  The area of the shaded region is half of the large circle minus half of one of the small circles. Note that the shaded half circle offsets one of the un shaded half circles.

  $\begin{array}{l}\text{A}\left(\text{shaded}\right)=\frac{1}{2}\text{A}\left(\text{lg}\text{circle}\right)-\frac{1}{2}\text{A}\left(\text{sm}\text{circle}\right)\\ =\frac{1}{2}\pi {\left(6\right)}^2-\frac{1}{2}\pi {\left(2\right)}^2\\ =18\pi -2\pi \\ \text{A}\left(\text{shaded}\right)=50.3{\text{in}}^2\end{array}$

Conclusion:

The area value is $\text{A}\left(\text{shaded}\right)=50.3{\text{in}}^2$