Question

find the arc - length of a circle with the given radius $r$ and central angle $\theta$. give the answer in the given unit of measure, rounded to the nearest hundredth.
$r = 31$ m; $\theta = 135^{circ}$

Explanation:

Step1: Convert angle to radians

First, convert $\theta = 135^{\circ}$ to radians. We know that $1^{\circ}=\frac{\pi}{180}$ radians. So $\theta=135\times\frac{\pi}{180}=\frac{3\pi}{4}$ radians.

Step2: Use arc - length formula

The arc - length formula is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius and $\theta$ is the central angle in radians. Given $r = 31$ m and $\theta=\frac{3\pi}{4}$, then $s=31\times\frac{3\pi}{4}=\frac{93\pi}{4}$.

Step3: Calculate the value

Calculate $\frac{93\pi}{4}\approx\frac{93\times3.14159}{4}=\frac{292.16787}{4}=73.0419675\approx73.04$ m.

Answer:

$73.04$ m