To find the length of the arc ([tex]\(s\)[/tex]) intercepted by a central angle ([tex]\(\theta\)[/tex]) on a circle with a given radius ([tex]\(r\)[/tex]), we follow these steps:
1. Convert the central angle [tex]\(\theta\)[/tex] from degrees to radians:
The formula for converting degrees to radians is:
[tex]\[
\text{Radians} = \theta \times \left( \frac{\pi}{180} \right)
\][/tex]
Given:
[tex]\[
\theta = 175^\circ
\][/tex]
So,
[tex]\[
\theta_{\text{radians}} = 175 \times \left( \frac{\pi}{180} \right) = \frac{175\pi}{180} = \frac{35\pi}{36} \text{ radians}
\][/tex]
2. Find the length of the arc [tex]\(s\)[/tex] using the formula [tex]\( s = r \theta_{\text{radians}} \)[/tex]:
Given:
[tex]\[
r = 16 \text{ inches}
\][/tex]
The arc length formula is:
[tex]\[
s = 16 \times \frac{35\pi}{36}
\][/tex]
Simplify the fraction:
[tex]\[
s = \frac{16 \times 35\pi}{36} = \frac{560\pi}{36} = \frac{140\pi}{9}
\][/tex]
Thus, the exact expression for the arc length [tex]\(s\)[/tex] in terms of [tex]\(\pi\)[/tex] is:
[tex]\[
s = \frac{140\pi}{9} \text{ inches}
\][/tex]
3. Round the arc length to two decimal places:
To round the arc length to two decimal places, we can evaluate the approximate value of [tex]\(\frac{140\pi}{9}\)[/tex] using [tex]\(\pi \approx 3.14159\)[/tex]:
[tex]\[
\frac{140\pi}{9} \approx \frac{140 \times 3.14159}{9} \approx 48.87 \text{ inches}
\][/tex]
So, the arc length [tex]\(s\)[/tex] rounded to two decimal places is:
[tex]\[
s \approx 48.87 \text{ inches}
\][/tex]
Hence, the exact answer in terms of [tex]\(\pi\)[/tex] is [tex]\(\frac{140\pi}{9}\)[/tex] inches, and the rounded answer to two decimal places is 48.87 inches.
