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 = 17 inches; central angle, θ = 95°. s = 323π/36 inches (simplify your answer. type an exact answer in terms of π. use integers or fractions for any numbers in the expression.) s = inches (round to two decimal places as needed.)

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s$ of a circle with radius $r$ and central angle $\theta$ (in degrees) is $s=\frac{\theta}{360}\times2\pi r$.

Step2: Substitute given values

Given $r = 17$ inches and $\theta=95^{\circ}$, we substitute into the formula:
\[

$$\begin{align*} s&=\frac{95}{360}\times2\pi\times17\\ &=\frac{95\times2\times17\pi}{360}\\ &=\frac{323\pi}{36}\text{ inches} \end{align*}$$

\]

Step3: Calculate the decimal value

To round to two decimal places, we calculate $\frac{323\pi}{36}$.
\[

$$\begin{align*} s&=\frac{323\times3.14159}{36}\\ &=\frac{1014.73357}{36}\\ &\approx28.19\text{ inches} \end{align*}$$

\]

Answer:

$28.19$