Question

find the length of the arc, s, on a circle of radius r. radius, r = 20 feet; central angle, θ = 200°. s = feet (simplify your answer. type an exact answer. s = feet (round to two decimal places as needed.

Explanation:

Step1: Convert angle to radians

First, convert the central - angle from degrees to radians. The conversion formula is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg} = 200^{\circ}$, then $\theta_{rad}=200\times\frac{\pi}{180}=\frac{10\pi}{9}$ radians.

Step2: Use arc - length formula

The arc - length formula is $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Given $r = 20$ feet and $\theta=\frac{10\pi}{9}$ radians, then $s=20\times\frac{10\pi}{9}=\frac{200\pi}{9}$ feet.

Step3: Calculate the decimal value

To get the value rounded to two decimal places, calculate $\frac{200\pi}{9}\approx\frac{200\times3.14159}{9}\approx69.81$ feet.

Answer:

$\frac{200\pi}{9}$ feet
$69.81$ feet