Question

if the radius is doubled, what is the effect on the area of sector xyz?
∘ the sector area will be \\(\frac{2}{7}\\) times greater.
∘ the sector area will be 2 times greater.
∘ the sector area will be \\(\frac{2}{7}\\) times greater.
∘ the sector area will be 4 times greater.

Explanation:

Step1: Recall the formula for the area of a sector

The area of a sector of a circle with radius \( r \) and central angle \( \theta \) (in radians) is given by \( A=\frac{1}{2}r^{2}\theta \).

Step2: Let the original radius be \( r \) and the new radius be \( R = 2r \)

First, find the original area \( A_1 \) with radius \( r \) and angle \( \theta=\frac{2\pi}{3} \):
\( A_1=\frac{1}{2}r^{2}\times\frac{2\pi}{3}=\frac{\pi r^{2}}{3} \)

Then, find the new area \( A_2 \) with radius \( R = 2r \) and the same angle \( \theta=\frac{2\pi}{3} \):
\( A_2=\frac{1}{2}(2r)^{2}\times\frac{2\pi}{3}=\frac{1}{2}\times4r^{2}\times\frac{2\pi}{3}=\frac{4\pi r^{2}}{3} \)

Step3: Find the ratio of the new area to the original area

Calculate \( \frac{A_2}{A_1} \):
\( \frac{A_2}{A_1}=\frac{\frac{4\pi r^{2}}{3}}{\frac{\pi r^{2}}{3}} = 4 \)
This means the new area is 4 times the original area.

Answer:

The sector area will be 4 times greater. (The option corresponding to this statement, e.g., if the last option is "The sector area will be 4 times greater", then the answer is that option. Assuming the options are as given, the correct option is the one stating the sector area will be 4 times greater.)