Question

Find the length of the arc indicated in bold. Leave your answer in terms of π. page 14 of 14 105 7 yd (1 point) • 17640 yards 40 π 12 yards Ο 14π yards 343 yards 24 < Find the length of the arc for a circle with a radius of 18 inches and a central angle of 300°. (1 point) Ο 324 inches Ο 30 inches Ο 270 inches O inches 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. (1 point) 44.0 mi 75.4 mi 452.4 mi

Answer 1

To find the length of an arc given the radius and the central angle in degrees, you can use the formula:

\[
L = \frac{\theta}{360} \times 2\pi r
\]

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

### Problem 1
**Circle with a radius of 18 inches and a central angle of 300°:**
- \(r = 18\)
- \(\theta = 300\)

Calculating the arc length:

\[
L = \frac{300}{360} \times 2\pi \times 18
\]

First, simplify \(\frac{300}{360}\):

\[
\frac{300}{360} = \frac{5}{6}
\]

Now, substitute and calculate:

\[
L = \frac{5}{6} \times 2\pi \times 18 = \frac{5}{6} \times 36\pi = 30\pi \text{ inches}
\]

### Problem 2
**Circle with a radius of 7 miles and a central angle of 210°:**
- \(r = 7\)
- \(\theta = 210\)

Calculating the arc length:

\[
L = \frac{210}{360} \times 2\pi \times 7
\]

First, simplify \(\frac{210}{360}\):

\[
\frac{210}{360} = \frac{7}{12}
\]

Now, substitute and calculate:

\[
L = \frac{7}{12} \times 2\pi \times 7 = \frac{7}{12} \times 14\pi = \frac{98\pi}{12} = \frac{49\pi}{6} \text{ miles}
\]

Now, to find the numerical value up to the tenths place, calculate:

\[
\frac{49\pi}{6} \approx \frac{49 \times 3.14}{6} \approx \frac{153.86}{6} \approx 25.64 \text{ miles}
\]

### Summary of Answers
1. Length of the arc for the first circle: **30π inches**
2. Length of the arc for the second circle, rounded to the tenths place: **25.6 miles**