A rear windshield wiper on a car has a total arm length of 10 inches and rotates back and forth through an angle of 95 degrees. The actual wiper blade is 7 inches long. Find the area of thr windshield that is being cleaned.
Take (95/360) of pi*[10^2 -3^2]
Do you see why?
Imagine a 10 inch radius circle with a concentric 3 inch radius circle taken out of the middle. The radial distance from the smaller to the larger radius is 7 inches. Your area is a 95 degree segment of that full circle.
To find the area of the windshield that is being cleaned, we need to calculate the area swept by the rear windshield wiper blade.
1. First, we need to find the length of the arc swept by the rear wiper blade. Since the wiper arm rotates back and forth through an angle of 95 degrees, we can calculate the length of the arc using the formula:
arc length = (angle / 360°) × circumference
The circumference of a circle can be calculated using the formula:
circumference = 2 × π × radius
Given that the total arm length is 10 inches, we can find the radius by dividing the total arm length by 2π:
radius = arm length / (2 × π)
Plugging in the values, we get:
radius = 10 / (2 × 3.1416) ≈ 1.592 inches
Now, we can calculate the arc length:
arc length = (95 / 360) × (2 × 3.1416 × 1.592)
≈ (95 / 360) × 9.996
≈ 2.637 inches
2. Next, we need to find the area of the windshield being cleaned by the wiper blade. This can be approximated by a sector of a circle. The area of a sector can be calculated using the formula:
sector area = (angle / 360°) × (π × radius^2)
Plugging in the values, we get:
sector area = (95 / 360) × (3.1416 × (1.592)^2)
≈ (95 / 360) × 7.968
≈ 2.114 square inches
Therefore, the approximate area of the windshield that is being cleaned by the rear wiper blade is 2.114 square inches.
To find the area of the windshield that is being cleaned by the wiper, we need to determine the area covered by the wiper blade as it moves through its range of motion.
First, let's determine the length of the arc created by the wiper blade. The formula to calculate the length of an arc is given by:
Arc Length = (θ/360) * (2π * r),
where θ is the central angle in degrees and r is the radius of the circle.
In this case, the central angle is 95 degrees, and the radius can be obtained by subtracting the length of the wiper blade from the total arm length:
Radius = Total arm length - Length of wiper blade
= 10 inches - 7 inches
= 3 inches.
Using the formula, we can calculate the length of the arc covered by the wiper blade:
Arc Length = (95/360) * (2π * 3)
= (19/72) * (6π)
= (19π/12) inches.
Now, let's find the area covered by the wiper blade. Since the wiper blade moves back and forth, the area is essentially a sector of a circle. The formula to calculate the area of a sector is given by:
Area of Sector = (θ/360) * (π * r^2).
Plugging in the values, we get:
Area of Sector = (95/360) * (π * 3^2)
= (19/72) * (9π)
= (19π/8) square inches.
Therefore, the area of the windshield that is being cleaned by the wiper blade is approximately (19π/8) square inches.