An automobile windshield washer 10inches long rotates an angle of 60degrres. If the rubber part of the blade only covers the last 9inches of the wiped, find the area of the windshield cleaned by the wiper.
The wiper covers 60 deg, or 1/6 of a circle. The area of a circle is πr^2. So, the area of a 10" circle is 100π. But we have to subtract the area of a 1" circle, since the rubber does not cover the inside inch. That would be π. So, a whole 10" circle minus a 1" circle would be 99π. But we only have 1/6 of that, or 16.5π = 51.8 sq. in.
To find the area of the windshield cleaned by the wiper, we need to determine the area covered by the rubber part of the blade.
First, let's assume that the wiper blade moves in a perfect arc across the windshield.
The total length of the wiper blade is 10 inches, but the rubber part only covers the last 9 inches. This means that when it rotates an angle of 60 degrees, the rubber part of the blade will also cover an arc length of 9 inches.
To find the radius of the circular arc, we can use the formula:
arc length = radius * angle in radians
We know that the arc length is 9 inches, and the angle is 60 degrees. To convert the angle to radians, we multiply it by π/180:
9 = r * (60 * π/180)
Simplifying, we have:
9 = r * (π/3)
Now, we can solve for the radius:
r = 9 / (π/3)
r = 27 / π
Finally, we can calculate the area of the windshield cleaned by the wiper.
The area of a circular sector is given by the formula:
area = (1/2) * r^2 * angle in radians
In this case, the angle is 60 degrees, which is equivalent to π/3 radians:
area = (1/2) * (27/π)^2 * (π/3)
Simplifying, we get:
area = 81/π square inches
Therefore, the area of the windshield cleaned by the wiper is approximately 25.8 square inches.
To find the area of the windshield cleaned by the wiper, we first need to calculate the length of the arc swept by the wiper blade.
The length of an arc can be calculated using the formula:
Length of arc = (θ/360) x (2πr)
Where θ is the angle in degrees, and r is the radius of the circle that the arc is a part of.
In this case, the length of the arc swept by the wiper blade is equal to 9 inches, since the rubber part of the blade only covers the last 9 inches of the wiped area. The angle swept by the blade is 60 degrees, and the radius of the circle that the arc is a part of is the length of the wiper blade, which is 10 inches.
So, let's calculate the length of the arc:
Length of arc = (60/360) x (2π x 10)
Simplifying the above equation, we get:
Length of arc = (1/6) x (20π) = (10/3)π
Now that we have the length of the arc, we can calculate the area of the windshield cleaned by the wiper. The cleaned area is essentially a sector of a circle with the same angle (60 degrees) and radius (10 inches).
The formula to find the area of a sector is:
Area of sector = (θ/360) x (πr^2)
So, the area of the windshield cleaned by the wiper is:
Area of sector = (60/360) x (π x 10^2) = (1/6) x (100π) = (50/3)π
Therefore, the area of the windshield cleaned by the wiper is (50/3)π square inches.