Question

use the following information to answer the next question.
the wiper on the rear window of a particular car moves through an angle of \\(\frac{5\pi}{6}\\) rad in a single sweep and clears a region that is 16 in wide, as shown in the diagram below.

1. the total perimeter of the cleared region, to the nearest inch, is

a. 52 in
b. 63 in
c. 84 in
d. 95 in

Explanation:

Step1: Identify two radii

The two radii of the sector are $r_1 = 16 +4 = 20$ in and $r_2 =4$ in.

Step2: Calculate arc lengths

Arc length formula: $s = r\theta$.
For $r_1$: $s_1 = 20\times\frac{5\pi}{6} = \frac{100\pi}{6} = \frac{50\pi}{3}$
For $r_2$: $s_2 = 4\times\frac{5\pi}{6} = \frac{20\pi}{6} = \frac{10\pi}{3}$

Step3: Add straight edges

Add two 16 in straight sides: $16 +16 =32$

Step4: Sum total perimeter

Total perimeter $P = \frac{50\pi}{3} + \frac{10\pi}{3} +32 = \frac{60\pi}{3} +32 = 20\pi +32$
$20\pi \approx 62.83$, so $62.83 +32 =94.83$

Answer:

D. 95 in