Question

An automobile windshield wiper 4 inches long rotates through an angle of 60 degrees . if the rubber part of the blade covers only the last 3 inches of the wiper, find the area of the windshield cleaned by the windshield wiper. answer exactly or round to 1 decimal place.

Answer 1

The area of the windshield cleaned by the wiper is approximately 7.07 square inches.

Step-by-step explanation:

Identify the relevant information:

Length of wiper (L) = 4 inches

Rotation angle (θ) = 60 degrees

Cleared area length (l) = 3 inches

Calculate the sector angle of the wiper's path:

The wiper covers only the last 3 inches of its length, so the sector angle (θ_s) is proportional to the ratio of the cleared length to the total length:

θ_s = θ * (l / L)

θ_s = 60° * (3 inches / 4 inches)

θ_s = 45°

Calculate the radius of the sector:

The radius (r) is equal to the length of the wiper:

r = L

r = 4 inches

Calculate the area of the sector:

A_s = (θ_s / 360°) * π * r^2

A_s = (45° / 360°) * π * (4 inches)^2

A_s ≈ 7.07 square inches

Therefore, the area of the windshield cleaned by the wiper is approximately 7.07 square inches.