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 = 12 feet; central angle, θ = 205° 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 the central - angle from degrees to radians. The conversion formula is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg} = 205^{\circ}$, then $\theta_{rad}=205\times\frac{\pi}{180}=\frac{41\pi}{36}$ radians.

Step2: Use the arc - length formula

The arc - length formula is $s = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central angle in radians. Given $r = 12$ feet and $\theta=\frac{41\pi}{36}$ radians, then $s=12\times\frac{41\pi}{36}$.

Step3: Simplify the expression

$12\times\frac{41\pi}{36}=\frac{41\pi}{3}$ feet (exact answer in terms of $\pi$).

Step4: Round the answer

To round to two decimal places, calculate the numerical value of $\frac{41\pi}{3}$. $\frac{41\pi}{3}\approx\frac{41\times3.14159}{3}\approx42.90$ feet.

Answer:

Exact answer: $\frac{41\pi}{3}$ feet
Rounded answer: $42.90$ feet