Question

the circle below has center o, and its radius is 7 m. given that m∠aob=30°, find the area of the shaded region and the length of the arc \\(\overarc{adb}\\). give exact answers in terms of π, and be sure to include the correct units in your answer.

Explanation:

Step1: Find the central angle of the shaded region

The total angle around a point is \(360^\circ\). Given \(\angle AOB = 30^\circ\), the central angle of the shaded region (\(\theta_{shaded}\)) is \(360^\circ - 30^\circ=330^\circ\). We can convert this to radians or use the formula for the area of a sector directly with degrees. The formula for the area of a sector with radius \(r\) and central angle \(\theta\) (in degrees) is \(A=\frac{\theta}{360^\circ}\times\pi r^{2}\). The formula for the length of an arc with radius \(r\) and central angle \(\theta\) (in degrees) is \(L = \frac{\theta}{360^\circ}\times2\pi r\).

Step2: Calculate the area of the shaded region

Given \(r = 7\space m\) and \(\theta = 330^\circ\). Substitute into the area formula:

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Step3: Calculate the length of arc \(\overarc{ADB}\)

Using the arc length formula with \(r = 7\space m\) and \(\theta = 330^\circ\):

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Answer:

Area of shaded region: \(\frac{539\pi}{12}\space m^{2}\)
Length of \(\overarc{ADB}\): \(\frac{77\pi}{6}\space m\)