Question

A rectangular window measures 54 inches by 24 inches. There is an a = 14-inch wiper blade attached by a b = 5-inch arm at the center of the base of the window, as shown in the figure below. If the arm rotates 120°, approximate the percentage of the window's area that is wiped by the blade.

Answer 1

It the wiper blade (and arm) made a full rotation, 2 circles would be formed.

One small circle with radius b, and a larger circle with radius a+b.


The areas of these 2 circles are respectively:

$\pi b^2=\pi\cdot 5^2=25 \pi (in^2)$    and

$\pi (a+b)^2=\pi (14+5)^2=361\pi (in^2)$


120° is 1/3 of a complete angle 360° which form a circle, so the areas formed by the arm of length b, and the arm + the wiper blade (a+b) are:

$\frac{25 \pi}{3}$ and $\frac{361\pi}{3}$ in squared respectively.

the actual wiped area is $\frac{361\pi}{3}-\frac{25 \pi}{3}= \frac{336\pi}{3}=112\pi \approx351.7$ (in squared)


Area of the window is 54*24= 1296 (in squared)

thus, the ratio of the wiped area to the whole area of the window is 351.7/1296=0.271

Converted to percentages, this is 27.1%


Answer: 27.1%