Chapter 11.3, Problem 47HP

To determine

The area of the shaded region of the circle.

Answer to Problem 47HP

It requires less steps, is faster, and there is a lower probability for error.

Explanation of Solution

Given information:

Describe two methods you could use to find the area of the shaded region of the circle. Which method do you think is more efficient? Explain your reasoning.

  

Formula used:

  $A=\frac{x}{360}\pi r^2$

Calculation:

Method 1: In finding the shaded area of the circle by subtracting x from 360° and using the resulting measure in the formula for the area of a sector. 

Let x = 120 and r = 10.

360 − 120 = 240

  $\begin{array}{l}A=\frac{x}{360}\pi r^2\\ =\frac{240}{360}\pi {\left(10\right)}^2\\ =\frac{2}{3}\pi \left(100\right)\\ A=209.4\end{array}$

Method 2: In calculating the shaded area by finding the area of the entire circle, finding the area of the un-shaded sector using the formula for the area of a sector, and subtracting the area of the un-shaded sector from the area of the entire circle. 

Let x = 120 and r = 10.

  $\begin{array}{l}A\left(circle\right)=\pi r^2\\ =\pi {\left(10\right)}^2\\ A\left(circle\right)=100\pi \end{array}$

  $\begin{array}{l}\text{A}\left(\text{sector}\right)=100\pi -\frac{100}{3}\pi \\ =\frac{300}{3}\pi -\frac{100}{3}\pi \\ =\frac{200}{3}\pi \\ \text{A}\left(\text{sector}\right)=209.4\end{array}$

The method in which you find the ratio of the area of a sector to the area of the whole circle is more efficient. It requires less steps, is faster, and there is a lower probability for error.

Conclusion:

It requires less steps, is faster, and there is a lower probability for error.