Chapter 11.3, Problem 49HP

To determine

The measure of a sector of that circle doubles.

Answer to Problem 49HP

If the arc length is doubled, the area would also be twice as large

Explanation of Solution

Given information:

If the radius of the circle is getting doubled then to determine the measure of sector of that circle double.

Formula used:

  $A=\pi r^2$

Calculation:

If the radius of the circle doubles, the area will not double.

  $\begin{array}{l}A=\pi r^2\\ =\pi {\left(3\right)}^2\\ A=9\pi \end{array}$

  $\begin{array}{l}A=\pi {\left(2r\right)}^2\\ =\pi {\left(6\right)}^2\\ A=36\pi \end{array}$

 If the radius of the circle doubles, the area will be four times as great $36\pi =4\left(9\pi \right)$

If the arc length of a sector is doubled, the area of the sector is doubled.

  $\begin{array}{l}A=\frac{x}{360}\pi r^2\\ =\frac{60}{360}\pi {\left(10\right)}^2\\ =\frac{1}{6}\pi \left(100\right)\\ A=\frac{100}{6}\pi \end{array}$

  $\begin{array}{l}A=\frac{2x}{360}\pi r^2\\ =\frac{120}{360}\pi {\left(10\right)}^2\\ =\frac{1}{3}\pi \left(100\right)\\ A=\frac{100}{3}\pi \end{array}$

  $\frac{100}{3}\pi =2\left[\frac{100}{6}\pi \right]$

Since the arc length is not raised to a power, if the arc length is doubled, the area would also be twice as large

Conclusion:

If the arc length is doubled, the area would also be twice as large