trigonometry question?

Question

An automobile windshield wiper 10 inches long rotates through an angle of 60 degrees. If the rubber part of the blade covers only the last 9 inches of the whiper, find the area of the windshield cleaned by the windshield wiper?

Answer 1

If you know the formula for the area of a circle and you know how many degrees are in a circle, you can answer this.

The problem is solved by breaking up with windshield wiper into 2 parts: the entire 10 inch wiper and its smaller 1 inch "base" (which contains no rubber). Each of these pieces swipes out a 60 degree chunk of circle (like one 10 inch "long" slice of pie and one 1 inch long slice). To get the area of the 9 inch region, simply subtract the area of the smaller 1 inch region from the area of the larger 10 inch region.

The area of a circle is pi*r^2, so the area of each 60 degree region is equal to (60/360)pi*r^2.

For a 10 inch radius, this comes to 52.36 square inches, and for a 1 inch radius, the area is 0.52 square inches. Thus, the area of the 9 inch swath is 52.36 - 0.52 = 51.84 square inches.

Answer 2

Forgive me if I don't run through the numbers, but here's how to do it.

Think of two circles, the outer circle and the inner non-rubber circle. The inner circle has a radius of 1in, the outer has a radius of 10in.

The way to get the area of the rubber wiped area is to take the area of the full circle swiped area and subtract the small circle swiped area from that.

The area of a circle is pi*r^2. So go pi*10^2 - pi*1^2. This will give you the area of the rubber swiped part if the wiper did a full 360 degree rotation. Since it's only doing 1/6 of that (60 degrees), times this answer by 1/6.

That should get you the area cleaned by the windshield wiper.

Hope this helps.

Answer 3

well, it's like this.that the entire wiper cover 60 deg, therefore the are that the wiper covers is

wiper area = 60/360 x pi x r^2 = 60/360 x pi x (10)^2

60/360 is actually because of the area formula is for the entire circle, which is 360 deg.

since the rubber only covers the last 9 in, therefore we find the area not covered by the wiper. which is only 1 in

area not covered by rubber = 60/360 x pi x 1^2

so to find area cleaned by the wiper, it's only:

area rubber = area wiper - area not covered by rubber

Answer 4

if you approach this logically, the solution would be:

to get the total area the wiper would've cleaned if it rotated completely and the entire length was covered by rubber (100pi square inches), subtract the area not covered by rubber (1pi square inches) and multiply by the actual fraction covered by the wiper (60/360 = 1/6).
the answer would be 16.5pi square inches.

Answer 5

1/6 of an annulus:
http://en.wikipedia.org/wiki/Annulus_%28mathematics%29