Question

A car's rear windshield wiper rotates 140°. The total length of the wiper mechanism is 24 inches and the length of the wiper blade is 13 inches. Find the area wiped by the wiper blade.

Answer 1

The area wiped by the wiper blade is
`177\pi`

Explanation:

Given:

car's rear windshield wiper rotates = 140°

Total length of the wiper mechanism = 24 inches

Length of the wiper blade = 13 inches

To Find:

The area wiped by the wiper blade = ?

Solution:

The length of the wiper mechanism ,
`r_1` = 24

The length of the wiper mechanism ,
`r_2`= 13

Now arc radius

`r_1-r_2`= 24-13 = 11

The wipe rotates 140 degrees

Step 1: converting degree into radians

`\theta = 140 * (\pi)/(180)`

`\theta =(7\pi)/(9)`

Step 2: Finding the area of the sector of the circle with radius
`r_1`

The area of the sector

=
`(1)/(2)r_1 ^2\theta`

=
`(1)/(2)24^2* (7\pi)/(9)`

=
`(576)/(2) * (7\pi)/(9)`

=
`(4032 \pi)/(18)`---------------------------(1)

Step 2: Finding the area of the sector with radius
`r_2`

=
`(1)/(2)r_1 ^2\theta`

=
`(1)/(2)11^2* (7\pi)/(9)`

=
`(121)/(2) * (7\pi)/(9)`

=
`(1183 \pi)/(18)`---------------------------(2)

Step 3: Finding the area wiped by the wiper

The area wiped by the wiper = the area of the sector of the circle with radius
`r_1` - the area of the sector with radius
`r_2`

From(1) and(2)

The area wiped by the wiper =
`(4032 \pi)/(18)` -
`(847\pi)/(18)`

The area wiped by the wiper =
`(3185\pi)/(18)`

The area wiped by the wiper =
`177\pi`