Question

example 3: find the central angle (in radians and degrees) that is formed by a radius of 12 ft and an arc length of 31 ft 31 = θ12 31/12 = θ 12/12 θ≈2.583

Explanation:

Step1: Recall arc - length formula

The formula for arc - length is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the central angle in radians. We are given $s = 31$ ft and $r = 12$ ft. Rearranging the formula for $\theta$, we get $\theta=\frac{s}{r}$.

Step2: Calculate the central angle in radians

Substitute $s = 31$ and $r = 12$ into the formula: $\theta=\frac{31}{12}\approx2.583$ radians.

Step3: Convert radians to degrees

Use the conversion factor $\theta_{degrees}=\theta_{radians}\times\frac{180^{\circ}}{\pi}$. So, $\theta=\frac{31}{12}\times\frac{180^{\circ}}{\pi}=\frac{31\times180^{\circ}}{12\pi}=\frac{5580^{\circ}}{12\pi}=\frac{465^{\circ}}{\pi}\approx147.9^{\circ}$.

Answer:

In radians: $\frac{31}{12}\approx2.583$ radians; In degrees: $\frac{465}{\pi}\approx147.9^{\circ}$