To find the length of an arc in a circle, you can use the formula:
\[ L = r \theta \]
where:
- \( L \) is the arc length,
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.
First, you need to convert the angle from degrees to radians. The conversion formula is:
\[ \theta \text{ (in radians)} = \theta \text{ (in degrees)} \times \frac{\pi}{180} \]
For a central angle of 210 degrees:
\[ \theta = 210 \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6} \]
Now, substitute the values into the arc length formula. The radius \( r = 7 \) miles, and now we have \( \theta = \frac{7\pi}{6} \):
\[ L = 7 \times \frac{7\pi}{6} = \frac{49\pi}{6} \]
Now, calculate \( \frac{49\pi}{6} \):
Using \( \pi \approx 3.14159 \):
\[ L \approx \frac{49 \times 3.14159}{6} \approx \frac{153.93851}{6} \approx 25.65642 \]
Rounding this to the tenths place gives:
\[ L \approx 25.7 \text{ miles} \]
Thus, the length of the arc is approximately 25.7 miles.
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