Find the area wiped by the wiper blade. (Round your answer to one decimal place.)

Question

A car's rear windshield wiper rotates 130°. The total length of the wiper mechanism is 22 inches and the length of the wiper blade is 13 inches. Find the area wiped by the wiper blade. (Round your answer to one decimal place.) 

___in^2

Answer 1

130/360 = 13/36

(13/36)pi(13)^2= 13(169)pi/36 =2197pi/36=about 2197(3.141593)/36=

191.7 in^2

(assuming the 22 inch numberis irrelevant and a trick distraction)

Answer 2

This solution assumes three things, which I think are reasonable assumptions:

1. The entire wiper remains on the surface to be wiped throughout its 130º rotation.
2. The wiper blade is located at the end of the wiper mechanism (as far away from the point of rotation as possible) as is common for all wipers I've seen.
3. The surface being wiped is flat and parallel to the plane of rotation of the wiper mechanism at every point under the area traced out by the wiper mechanism.

It might be helpful to draw out a picture of the scenario; you can see mine here:

https://imgur.com/TAdflKg

To calculate the shaded area labeled "wiped surface", imagine that the wiper mechanism doesn't stop at 130º, but instead goes all 360º around. Then it's easy to see that the wiper mechanism traces out an area of a circle ("Circle A" = π • r²), where r = 22". However, the blade is located at the end of the mechanism and only wipes the surface between 9" and 22" away from the point of rotation (the center of the circles A and B); we arrive at 9" by subtracting the length of the blade from the total length of the mechanism (22" - 13" = 9"). The area between 0 to 9 inches from the point of rotation (area of "Circle B") is not wiped. Thus, to calculate the area that is wiped, we subtract the area of the inner circle ("Circle B") from the total area traced out by the whole mechanism for a 360º rotation ("Circle A"):

Α_wiped(360º) = A_circleA - A_circleB = π • 22² - π • 9² = π • (484 - 81) = π • 403

The last thing to do is to remember that we aren't actually rotating the mechanism through a full circle, but only through 130º of 360º of a circle. So, we multiply our last step by the proportion 130º / 360º = 13 / 36. For our final answer then...

Α_wiped(130º) = (13 / 36) • π • 403 = 457.18899755…

Remembering that our units will be square inches (in²) and rounding to a single decimal place:

Α_wiped(360º) = 457.2 in²