Why Subtract Non-Cleaning Area in Windshield Wiper Sector Calculations?

[theintarnets]

Homework Statement

The total arm and blade of a single windshield wiper was 10 in. long and rotated back and forth through an angle of 95°. A portion of the windshield is cleaned by 7-in. wiper blade. What is the area of the region cleaned?

Homework Equations

Area of a sector = 1/2 r²θ

The Attempt at a Solution

My first guess was to simply calculate the area cleaned by the blade and I thought that would be my answer.

In the book though, they calculated the area not cleaned by the blade (r would be equal to (10-7)²) and subtracted it from the total area. I can't wrap my head around that. Why subtract the two areas? Why not just calculate the area that was cleaned by the blade? My guess is it has something to do with the actual angle or radius, but I'd like a little clarification to make sure I really understand this. Thanks!

 

[tiny-tim]

hi theintarnets!

there is a difference between θ/2 * (10² - 7²) and θ/2 * (10 - 7)² …

which one is correct?​

 

[theintarnets]

They were using the portion not cleaned by the blade which was equal to 3 in., so (10-7) was used for r, and then squared.

 

[tiny-tim]

so they used θ/2 * (10² - 3²) ?

what do you think is wrong with that?

what do you think it should be?

 

[Joffan]

The Attempt at a Solution

My first guess was to simply calculate the area cleaned by the blade and I thought that would be my answer.

OK, well, that's what the question asks for. Show us how you would do that.

Imagine a worn-out wiper arm, total length 10", that only has 1" of cleaning blade at the end. How would that calculation look different?

 

[theintarnets]

What I had done was just use 7 for r, thinking that since the cleaning blade was 7 inches, the area cleaned would be 1/2*(7)^2*whatever 95 degrees in radians is. But they way you put it does make me think about it a little more. I guess the area cleaned by blade 1 inch long is much more different than the area cleaned by the last inch of a 10 inch blade. So then the only way to figure it out would be to get the whole area, then subtract the area that the blade isn't cleaning. Am I right?

 

[tiny-tim]

yup!

 

[HallsofIvy]

Did you draw a picture? One thing you need to know, that you did not state, is that the 7" 'cleaning' part of the wiper is at the end of the 10" wiper. IF the 'cleaning' part had started at the pivot point, then you would only need the 7" part. Of course, anyone who has looked at a windshield wiper knows the 'cleaning' part is at the end opposite the wiper.