Question

the radius of a circle is 9 inches. what is the length of a 150° arc?
give the exact answer in simplest form.

Explanation:

Step1: Recall arc length formula

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

Step2: Substitute given values

Here, \( \theta = 150^\circ \) and \( r = 9 \) inches. Substitute these values into the formula:
\( L=\frac{150^\circ}{360^\circ}\times2\pi\times9 \)

Step3: Simplify the expression

First, simplify \( \frac{150}{360}=\frac{5}{12} \), and \( 2\times9 = 18 \). Then we have:
\( L=\frac{5}{12}\times18\pi \)
Simplify \( \frac{5}{12}\times18=\frac{5\times18}{12}=\frac{90}{12}=\frac{15}{2} \). So \( L = \frac{15}{2}\pi \) or \( 7.5\pi \). But in exact form, it's \( \frac{15}{2}\pi \).

Answer:

\( \frac{15}{2}\pi \) (or \( 7.5\pi \)) inches