Question

the radius of a circle is 9 inches. what is the length of a 102° arc?
give the exact answer in simplest form.
inches

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) of a circle is \(s=\frac{\theta}{360}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.

Step2: Substitute the given values

Given that \(r = 9\) inches and \(\theta=102^{\circ}\).
Substitute into the formula: \(s=\frac{102}{360}\times2\pi\times9\).
First, simplify \(\frac{102}{360}=\frac{17}{60}\).
Then, \(s=\frac{17}{60}\times2\pi\times9\).
\(s=\frac{17\times2\times9\pi}{60}\).
\(2\times9 = 18\), so \(s=\frac{17\times18\pi}{60}\).
\(17\times18=306\), then \(s=\frac{306\pi}{60}\).
Simplify the fraction \(\frac{306}{60}=\frac{51}{10}\).

Answer:

\(\frac{51}{10}\pi\) inches