Question

find the length of the arc, s, on a circle of radius r intercepted by a central angle θ. express arc length in terms of π. then round your answer to two decimal places. radius, r = 19 feet; central angle, θ = 260°. s = feet (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 $\theta = 260^{\circ}$ to radians. We know that $1^{\circ}=\frac{\pi}{180}$ radians. So $\theta=260\times\frac{\pi}{180}=\frac{13\pi}{9}$ radians.

Step2: Use 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 $r = 19$ feet and $\theta=\frac{13\pi}{9}$ radians, then $s=19\times\frac{13\pi}{9}=\frac{247\pi}{9}$ feet.

Step3: Round the answer

To round to two decimal places, calculate the numerical value of $\frac{247\pi}{9}$. $\frac{247\pi}{9}\approx\frac{247\times 3.14159}{9}\approx\frac{775.97273}{9}\approx86.22$ feet.

Answer:

Exact answer: $\frac{247\pi}{9}$ feet
Rounded answer: $86.22$ feet