To find the length of the arc created by a central angle in a circle, we can use the formula for the arc length \(L\):
\[ L = r \theta \]
where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.
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Find the radius:
The diameter of the circle is given as 94 kilometers. The radius \(r\) is half of the diameter:
\[ r = \frac{d}{2} = \frac{94}{2} = 47 \text{ kilometers} \]
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Identify the central angle:
The central angle is given as \(\frac{3\pi}{2}\) radians.
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Calculate the arc length:
Substitute \(r\) and \(\theta\) into the arc length formula:
\[ L = r \theta = 47 \times \frac{3\pi}{2} \]
Now calculate the value:
\[ L = 47 \times \frac{3\pi}{2} = \frac{141\pi}{2} \]
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Approximate using \(\pi \approx 3.14\) (if required):
If you want a numerical approximation:
\[ L \approx \frac{141 \times 3.14}{2} \approx \frac{442.14}{2} \approx 221.07 \text{ kilometers} \]
Thus, the exact length of the arc created by the angle is
\[ \frac{141\pi}{2} \text{ kilometers} \]
If an approximate decimal value is needed, it is approximately:
\[ 221.07 \text{ kilometers} \]
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