Question

finding arc length in exercises 51 and 52, find the length of the arc on a circle of radius r intercepted by a central angle θ. 51. r = 15 inches, θ = 120°. 52. r = 3 meters, θ = 150°.

Explanation:

Step1: Convert angle to radians

The formula to convert degrees to radians is $\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}$. For $\theta = 120^{\circ}$ in problem 51, $\theta_{rad}=\frac{\pi}{180}\times120=\frac{2\pi}{3}$ radians. For $\theta = 150^{\circ}$ in problem 52, $\theta_{rad}=\frac{\pi}{180}\times150=\frac{5\pi}{6}$ 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).

For problem 51:

Given $r = 15$ inches and $\theta=\frac{2\pi}{3}$ radians, then $s=15\times\frac{2\pi}{3}=10\pi\approx 31.4$ inches.

For problem 52:

Given $r = 3$ meters and $\theta=\frac{5\pi}{6}$ radians, then $s = 3\times\frac{5\pi}{6}=\frac{5\pi}{2}\approx 7.85$ meters.

Answer:

Problem 51: The arc length is approximately $31.4$ inches.
Problem 52: The arc length is approximately $7.85$ meters.