Question

Find the length of the arc of a circle with a radius of 7 mi, and a central angle of 210°. Round your answer to the tenths place

Answer 1

To find the length of an arc of a circle, we can use the formula:

\[
L = r \theta
\]

where:
- \(L\) is the arc length,
- \(r\) is the radius of the circle, and
- \(\theta\) is the central angle in radians.

First, we need to convert the central angle from degrees to radians. The conversion from degrees to radians is given by the formula:

\[
\theta \text{ (radians)} = \theta \text{ (degrees)} \times \frac{\pi}{180}
\]

In your case, the central angle is \(210^\circ\):

\[
\theta = 210 \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6} \text{ radians}
\]

Now that we have \(\theta\) in radians, we can substitute the radius \(r = 7\) miles and \(\theta = \frac{7\pi}{6}\) into the arc length formula:

\[
L = 7 \times \frac{7\pi}{6}
\]

Calculating this gives:

\[
L = \frac{49\pi}{6}
\]

Next, we can use the approximate value of \(\pi \approx 3.14159\):

\[
L \approx \frac{49 \times 3.14159}{6} \approx \frac{153.9381}{6} \approx 25.65635
\]

Rounding to the tenths place, we find:

\[
L \approx 25.7 \text{ miles}
\]

Thus, the length of the arc is approximately **25.7 miles**.