Question

what is the sector area created by the hands of a clock with a radius of 9 inches when the time is 4:00? 6.75π in.² 20.25π in.² 27π in.² 81π in.²

Explanation:

Step1: Calculate the central angle

A clock is a circle, which has \(360^{\circ}\). At 4:00, the minute hand is at 12 and the hour hand is at 4. The number of intervals between 12 and 4 is 4. Each interval represents \(\frac{360^{\circ}}{12}=30^{\circ}\). So the central angle \(\theta = 4\times30^{\circ}=120^{\circ}\). Convert \(120^{\circ}\) to radians: \(\theta=\frac{120\pi}{180}=\frac{2\pi}{3}\) radians. The formula for the area of a sector is \(A = \frac{1}{2}r^{2}\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians). Given \(r = 9\) inches.

Step2: Substitute values into the formula

Substitute \(r = 9\) and \(\theta=\frac{2\pi}{3}\) into \(A=\frac{1}{2}r^{2}\theta\).

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Answer:

\(27\pi\) in.\(^{2}\) (the third option)