Question

if the radius is doubled, what is the effect on the area of sector xyz?
○ the sector area will be $\frac{3}{2}$ times greater.
○ the sector area will be 2 times greater.
○ the sector area will be $\frac{2}{3}$ times greater.
○ the sector area will be 4 times greater.

Explanation:

Step1: Recall sector area formula

The area of a sector with radius $r$ and central angle $\theta$ is $A = \frac{1}{2}r^2\theta$.

Step2: Define original sector area

For the given sector, $\theta = \frac{2\pi}{3}$, so original area:
$A_1 = \frac{1}{2}r^2 \cdot \frac{2\pi}{3} = \frac{\pi r^2}{3}$

Step3: Calculate new area (radius doubled)

New radius $r_2 = 2r$, so new area:
$A_2 = \frac{1}{2}(2r)^2 \cdot \frac{2\pi}{3} = \frac{1}{2}(4r^2) \cdot \frac{2\pi}{3} = \frac{4\pi r^2}{3}$

Step4: Find ratio of new to original area

$\frac{A_2}{A_1} = \frac{\frac{4\pi r^2}{3}}{\frac{\pi r^2}{3}} = 4$

Answer:

The sector area will be 4 times greater.