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.
area of a circle =pi(r)^2
since its 95 degrees 95/180=.5278
r=7 inches
therefore the area is 81.25
Yea, but it isn't the entire circle so a circle area formula won't work. The equation for this area will be A= (1/2)r*Given Angle...
To find the area of the windshield that is being cleaned, we can use the concept of a sector of a circle.
1. First, let's find the radius of the circle formed by the rear windshield wiper arm.
The total arm length is 10 inches, and it rotates back and forth through an angle of 95 degrees.
Since the arm is like a radius of the circle, we can use the formula:
arc length = radius x angle in radians
We need to convert the angle from degrees to radians:
angle in radians = (95 degrees) * (π/180)
Plugging in the values, we have:
10 inches = radius x (95 degrees) * (π/180)
Solve for the radius:
radius = (10 inches) / ((95 degrees) * (π/180))
2. Once we have the radius, we can calculate the area of the sector formed by the wiper blade.
The area of a sector can be calculated using the formula:
area of sector = (1/2) x radius^2 x angle in radians
Plugging in the values, we have:
area of sector = (1/2) x (radius^2) x (95 degrees) * (π/180)
Simplifying further:
area of sector = (1/2) x (radius^2) x ((95 degrees) * (π/180))
3. Now, let's find the area of the windshield that is being cleaned.
The wiper blade itself is 7 inches long, and it forms the chord of the sector.
We can find the area of the triangle formed by the chord and the two radii using the formula:
area of triangle = (1/2) x wiper blade length x radius
Plugging in the values, we have:
area of triangle = (1/2) x (7 inches) x radius
4. Finally, we can calculate the area of the cleaned windshield by subtracting the area of the triangle from the area of the sector:
area of cleaned windshield = area of sector - area of triangle
Plugging in the values, we have:
area of cleaned windshield = [(1/2) x (radius^2) x ((95 degrees) * (π/180))] - [(1/2) x (7 inches) x radius]
Calculate this expression to find the area of the windshield that is being cleaned.
To find the area of the windshield that is being cleaned by the rear windshield wiper, we need to calculate the area covered by the wiper blade as it moves back and forth. This area can be approximated as a sector of a circle.
First, let's calculate the length of the arc covered by the wiper blade. Since the total arm length is 10 inches and it rotates through an angle of 95 degrees, we can use the formula for arc length:
Arc length = (angle in degrees / 360) x (2πr)
where r is the radius of the circle. In this case, the radius is equal to the length of the wiper blade, which is 7 inches.
Arc length = (95 / 360) x (2π x 7)
Arc length ≈ 3.878 inches
Next, let's find the area of the sector formed by this arc. The formula to calculate the area of a sector is:
Area of sector = (angle in degrees / 360) x πr²
Area of sector = (95 / 360) x π x 7²
Area of sector ≈ 48.87 square inches
Therefore, the area of the windshield that is cleaned by the rear wiper blade is approximately 48.87 square inches.