Question

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what is the measure of the central angle of a circle with radius 60 centimeters that intercepts a 15π centimeters arc?
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Explanation:

Step1: Recall arc length formula

The formula for the length of an arc \( s \) is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians.

Step2: Solve for \( \theta \)

We know \( s = 15\pi \) and \( r = 60 \). Substitute into the formula: \( 15\pi = 60\theta \).
Divide both sides by 60: \( \theta=\frac{15\pi}{60}=\frac{\pi}{4} \) radians. To convert to degrees, use \( \theta_{degrees}=\theta_{radians}\times\frac{180^\circ}{\pi} \). So \( \frac{\pi}{4}\times\frac{180^\circ}{\pi} = 45^\circ \). Or, if we keep it in radians, \( \frac{\pi}{4} \) radians is also correct, but likely degrees is expected here. Wait, let's check the calculation again. Wait, \( 15\pi = 60\theta \), so \( \theta=\frac{15\pi}{60}=\frac{\pi}{4} \) radians, which is \( 45^\circ \). Wait, no, wait: \( s = r\theta \), so \( \theta=\frac{s}{r} \). So \( \theta=\frac{15\pi}{60}=\frac{\pi}{4} \) radians, which is 45 degrees. Wait, but let's confirm. Alternatively, maybe the question expects the answer in radians or degrees? Let's see, the arc length is \( 15\pi \), radius 60. So \( \theta=\frac{15\pi}{60}=\frac{\pi}{4} \) radians, which is 45 degrees. Wait, but let's check the arithmetic. 15 divided by 60 is 0.25, so \( 0.25\pi \) radians, which is \( \frac{\pi}{4} \) radians, or 45 degrees. So the central angle is \( \frac{\pi}{4} \) radians (45 degrees). Wait, but maybe the problem is expecting the answer in degrees? Let's see, if we calculate in degrees: \( \theta=\frac{s}{r}\times\frac{180^\circ}{\pi}=\frac{15\pi}{60}\times\frac{180^\circ}{\pi}=\frac{15\times180^\circ}{60}=45^\circ \). Yes, that's correct. So the central angle is 45 degrees (or \( \frac{\pi}{4} \) radians). But likely degrees is the answer here. Wait, but let's check the problem again. The problem says "measure of the central angle". So either radians or degrees, but since the arc length has \( \pi \), maybe radians? Wait, no, \( \frac{\pi}{4} \) radians is 45 degrees. Wait, maybe the answer is 45 degrees (or \( \frac{\pi}{4} \) radians). Let's confirm the calculation: \( s = r\theta \), so \( \theta = s/r = 15\pi / 60 = \pi/4 \) radians, which is 45 degrees. So the central angle is 45 degrees (or \( \pi/4 \) radians).

Answer:

\( 45^\circ \) (or \( \frac{\pi}{4} \) radians, but 45 degrees is more likely the expected answer here)