Question

find the area of a sector of a circle having radius r and central angle θ.
r = 20.0 mi, θ = 145°
the area is approximately □ mi².
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Convert the angle from degrees to radians

The formula for the area of a sector is \(A=\frac{1}{2}r^{2}\theta\) (where \(\theta\) is in radians).
We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees).
So, \(\theta = 145^{\circ}\times\frac{\pi}{180}=\frac{29\pi}{36}\) radians.

Step2: Calculate the area of the sector

Given \(r = 20.0\) mi.
Substitute \(r = 20\) and \(\theta=\frac{29\pi}{36}\) into the formula \(A=\frac{1}{2}r^{2}\theta\).
\(A=\frac{1}{2}\times(20)^{2}\times\frac{29\pi}{36}\)
\(=\frac{1}{2}\times400\times\frac{29\pi}{36}\)
\( = 200\times\frac{29\pi}{36}\)
\(=\frac{5800\pi}{36}\)
\(=\frac{1450\pi}{9}\approx\frac{1450\times 3.14159}{9}\)
\(=\frac{4555.3055}{9}\approx506.1\)

Answer:

\(506.1\)