Question

An arc of length 15 ft subtends a central angle$\theta$ in a circle of radius 9 ft .find the measure of$\theta$ in degrees and radians.

Short answer

The central angle$\theta$ is $\mathbf{95}{\mathbf{.54}}^{\mathbf{\circ }}\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\frac{\mathbf{5}}{\mathbf{3}}\text{radian}$.

Step-by-step solution

Step 1. Given information

The given radius of $\text{circle}=9$ft and the length of the arc is 15 ft.

Step 2. Explanation

We will use the formula of arc length,

$s=r\theta$

Step 3. Calculation

$\begin{array}{l}\text{Thelengthof\hspace{0.17em}the\hspace{0.17em}arc\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}s=r\theta \\ \theta =\frac{s}{r}\\ \theta =\frac{15}{9}\\ \theta =\frac{5}{3}\text{\hspace{0.17em}radian}\end{array}$

Now to convert central angle in degrees, multiply by $\frac{180}{\pi }$

$\begin{array}{l}\theta =\frac{5}{3}.\frac{180}{\pi }\\ \theta =\frac{300}{\pi }\\ \theta ={95.54}^{\circ }\end{array}$