To find the length of the arc created by a central angle in a circle, you can use the formula:
\[ \text{Arc Length} = r \cdot \theta \]
where \( r \) is the radius of the circle, and \( \theta \) is the angle in radians.
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Calculate the radius: Given that the diameter of the circle is 30 inches: \[ r = \frac{\text{diameter}}{2} = \frac{30}{2} = 15 \text{ inches} \]
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Use the given central angle: The angle \( \theta \) is given as \( \frac{7\pi}{4} \) radians.
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Substitute the values into the formula: \[ \text{Arc Length} = r \cdot \theta = 15 \cdot \frac{7\pi}{4} \]
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Calculate the arc length: \[ \text{Arc Length} = 15 \cdot \frac{7\pi}{4} = \frac{105\pi}{4} \]
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Now compute \( \frac{105\pi}{4} \) numerically (using \( \pi \approx 3.14 \)): \[ \frac{105 \cdot 3.14}{4} \approx \frac{329.7}{4} \approx 82.425 \text{ inches} \]
Thus, the arc length is approximately 82.5 inches.
The answer is 82.5 in.