a windshield wiper on a a plymouth acclaim is 20 in. long and has a blade of 16in long. if the wiper sweeps through an angle of 110 degree, how large an area does the wiper blade clean?
a sector of angle θ in a circle of radius r is 1/2 r^2 θ
110° = 110/180 pi = 1.92
20 * 1.92^2 = 38.4
(20-16)* 1.92^2 = 7.7
38.4-7.7 = 30.7 in^2
Well, the wiper blade may not be the best at cleaning up crimes, but it sure does a good job cleaning windshields! Now, let's crunch some numbers.
To find the area that the wiper blade cleans, we need to know the length of the arc it sweeps. Given that the length of the blade is 16 inches and it sweeps through an angle of 110 degrees, we can calculate the arc length using the formula:
Arc Length = (angle/360) * (2 * π * radius)
Since the radius of the arc is half the length of the blade (16/2 = 8 inches), we can plug the values into the formula:
Arc Length = (110/360) * (2 * π * 8)
Simplifying further, we get:
Arc Length = (11/36) * (16 * π)
Arc Length ≈ 14.082 inches
So, the wiper blade cleans an area of approximately 14.082 square inches on the windshield. With that much coverage, your windshield will be looking spick and span!
To find the area that the wiper blade cleans, we need to calculate the swept area by multiplying the length of the wiper blade with the length of the arc it sweeps.
1. First, we need to convert the degree measure to radians. We know that 360 degrees is equivalent to 2π radians, so we can set up a proportion to find the conversion factor:
360 degrees = 2π radians
110 degrees = x radians
Solving for x, we get:
x = (110 * 2π) / 360
≈ 1.92 radians
2. The length of the arc swept by the wiper blade can be calculated using the formula:
arc length = radius of the circle * angle in radians
Since the wiper sweeps through an angle of 110 degrees, we can use the radius of the windshield wiper as the radius of the circle. Therefore, the length of the arc is:
arc length = 20 in * 1.92 radians
≈ 38.4 in
3. Finally, we can calculate the area that the wiper blade cleans by multiplying the length of the arc by the length of the wiper blade:
area = arc length * wiper blade length
= 38.4 in * 16 in
= 614.4 square inches
Therefore, the wiper blade cleans an area of approximately 614.4 square inches.
To find the area cleaned by the wiper blade, we will consider it as a part of a circle. The length of the wiper blade acts as a radius, and the angle through which it sweeps forms a sector of the circle.
Let's break down the steps to find the area:
Step 1: Convert the degree measure to radians.
To find the area of the sector, we need the angle in radians. We know that π radians is equivalent to 180 degrees. Therefore, we can use the conversion factor:
angle in radians = angle in degrees * (π / 180)
In this case, the angle in degrees is 110. Applying the conversion, we get:
angle in radians = 110 * (π / 180)
Step 2: Calculate the length of the arc.
The arc length is the distance traveled by the wiper blade, which is equivalent to the length of the wiper blade itself. In this case, the length of the arc is 16 inches.
Step 3: Calculate the radius.
The length of the wiper is given as 20 inches, which acts as the radius of the circle.
Step 4: Calculate the area of the sector.
The formula to calculate the area of a sector is:
Area = (Angle in radians / 2π) * πr²
In this case, we have the angle in radians (calculated in Step 1) and the radius (calculated in Step 3). Plugging in the values, we get:
Area = (angle in radians / 2π) * π * r²
Step 5: Simplify and calculate.
By simplifying the formula, we get:
Area = (angle in radians / 2) * r²
Now, substitute the values:
Area = (110 * (π / 180) / 2) * 20²
Simplify further:
Area = 55 * (π / 180) * 400
Now, calculate:
Area = 55 * (π / 180) * 400 ≈ 383.98 square inches
Therefore, the wiper blade cleans an area of approximately 383.98 square inches.