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 the formula for the area of a sector

The formula for the area of a sector is \(A=\frac{1}{2}\theta r^{2}\), where \(\theta\) is the central - angle in radians and \(r\) is the radius. Given \(\theta = \frac{2\pi}{3}\) (constant in this case). Let the original radius be \(r\), so the original area \(A_1=\frac{1}{2}\times\frac{2\pi}{3}\times r^{2}=\frac{\pi r^{2}}{3}\).

Step2: Calculate the area when the radius is doubled

When the radius is doubled (\(r' = 2r\)), the new area \(A_2=\frac{1}{2}\times\frac{2\pi}{3}\times(2r)^{2}\).

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

\(\frac{A_2}{A_1}=\frac{\frac{4\pi r^{2}}{3}}{\frac{\pi r^{2}}{3}}\)
Since \(\frac{\frac{4\pi r^{2}}{3}}{\frac{\pi r^{2}}{3}} = 4\) (the \(\pi r^{2}/3\) terms cancel out).

Answer:

The sector area will be 4 times greater.