Question

find the length of the arc, s, on a circle of radius r intercepted by a central angle θ. express the arc length in terms of π. then round your answer to two decimal places. radius, r = 16 inches; central angle, θ = 25°. s = □ inches (simplify your answer. type an exact answer in terms of π. use integers or fractions for any numbers in the expression.)

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 $\theta_{deg} = 25^{\circ}$, then $\theta_{rad}=25\times\frac{\pi}{180}=\frac{5\pi}{36}$.

Step2: Use the arc - length formula

The formula for the arc - length $s$ of a circle is $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Given $r = 16$ inches and $\theta=\frac{5\pi}{36}$, then $s=16\times\frac{5\pi}{36}=\frac{20\pi}{9}$ inches.

Step3: Round the answer

To round to two decimal places, calculate the numerical value of $\frac{20\pi}{9}$. $\frac{20\pi}{9}\approx\frac{20\times3.14159}{9}\approx6.98$ inches.

Answer:

Exact answer: $\frac{20\pi}{9}$ inches
Rounded answer: $6.98$ inches