Question

find the length of the arc of a circle of radius 12 inches subtended by a central angle of 110°. answer exactly in terms of pi. inches question help: video written example message instructor submit question

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) of a circle is \(s = r\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians. First, convert the angle from degrees to radians. We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). Given \(\theta = 110^{\circ}\), then \(\theta=\frac{110\pi}{180}=\frac{11\pi}{18}\) radians.

Step2: Substitute the values into the arc - length formula

We are given \(r = 12\) inches and \(\theta=\frac{11\pi}{18}\) radians. Using the formula \(s=r\theta\), we substitute the values: \(s = 12\times\frac{11\pi}{18}\).

Step3: Simplify the expression

Simplify \(12\times\frac{11\pi}{18}\). We can cancel the common factor of 6. \(12=\ 2\times6\) and \(18 = 3\times6\). So \(12\times\frac{11\pi}{18}=\frac{2\times6\times11\pi}{3\times6}=\frac{2\times11\pi}{3}=\frac{22\pi}{3}\)

Answer:

\(\frac{22\pi}{3}\)