24°
To determine the angle described by the minute hand during the period when it sweeps an area of \(15\pi \text{ cm}^2\), we need to use the formula for the area swept by the minute hand of a clock:
The area \((A)\) swept by a sector of a circle is given by:
\(A = \frac{1}{2} r^2 \theta\)
Where:
- \(r\) is the radius (or length of the minute hand, which is \(15 \text{ cm}\)).
- \(\theta\) is the angle in radians swept by the hand.
Given that \(A = 15\pi \text{ cm}^2\), we can substitute these values into the formula:
\(\frac{1}{2} \times (15)^2 \times \theta = 15\pi\)
Solve for \(\theta\):
\(\frac{1}{2} \times 225 \times \theta = 15\pi\)
\(112.5 \theta = 15\pi\)
\(\theta = \frac{15\pi}{112.5}\)
\(\theta = \frac{\pi}{7.5} \text{ radians}\)
To convert the angle from radians to degrees, we use the conversion factor \(180^\circ = \pi \text{ radians}\):
\(\theta = \frac{\pi}{7.5} \times \frac{180}{\pi}\)
\(\theta = \frac{180}{7.5}\)
\(\theta = 24^\circ\)
Therefore, the angle described by the minute hand is 24°.
Conclusion: The correct answer is 24°, which matches the option provided.
Similar Questions
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Take $\pi = \frac{22}{7}$
