To find the arc length created by a 60-degree central angle in a circle with a 10-inch diameter, we'll use the relationship you provided and perform the necessary conversions and calculations step by step.
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Convert the angle from degrees to radians. The conversion factor is that \(360\) degrees equal \(2\pi\) radians. To convert \(60\) degrees to radians: \[ \text{Radians} = 60 \times \left(\frac{2\pi}{360}\right) = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \]
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Determine the radius of the circle. The radius \(r\) is half of the diameter. Given the diameter is \(10\) inches: \[ r = \frac{10}{2} = 5 \text{ inches} \]
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Use the formula for arc length. The formula for arc length \(s\) when you have the central angle in radians is given by: \[ s = r\theta \] where \(r\) is the radius and \(\theta\) is the angle in radians. Substituting in the known values: \[ s = 5 \times \frac{\pi}{3} = \frac{5\pi}{3} \text{ inches} \]
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Conclude Thus, the length of the arc created by a 60-degree central angle in a circle with a 10-inch diameter is \( \frac{5\pi}{3} \) inches.
From the answer choices provided, the correct response that matches our calculation is: The answer is \( \frac{5\pi}{3} \) because we plugged in the known angle of 60 degrees in terms of radians by taking \( \frac{60}{360} \), or simplified to \( \frac{1}{6} \) of \( 2\pi \) to get \( \frac{\pi}{3} \) as the central angle in terms of radians. Next, we use the known proportional relationship replacing the angle and radius of 5 as it is half of the diameter given to get \( \frac{\pi}{3} = \frac{s}{5} \). By multiplying both sides by 5 we solve for our arc length and get \( s = \frac{5\pi}{3} \).