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 = 41π/3 feet (simplify your answer. type an exact answer in terms of π. use integers or fractions for any numbers in the expression.) s = feet (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 $r = 12$ feet and $\theta=205^{\circ}$, we substitute into the formula: $s=\frac{205}{360}\times2\pi\times12$.
First, simplify $\frac{205}{360}\times2\times12$:
$\frac{205}{360}\times24=\frac{205\times24}{360}=\frac{4920}{360}=\frac{41}{3}$.
So, $s = \frac{41\pi}{3}$ feet (exact answer).

Step3: Calculate the approximate value

To find the approximate value, we know that $\pi\approx3.14159$.
$s=\frac{41\times3.14159}{3}$.
$41\times3.14159 = 128.80519$.
Then $\frac{128.80519}{3}\approx42.94$ feet.

Answer:

42.94