A windshield wiper on a certain model car pivots through an angle of 123°. The total arm and blade length is 24 inches. If the blade is 21 inches, what is the total area of windshield being cleaned by the wiper?
area of a sector is
a = 1/2 r^2 θ
so, the area is
1/2 π(123/180) (24^2-3^2) = 608.6 in^2
To find the total area of the windshield being cleaned by the wiper, we need to calculate the area covered by the wiper blade as it pivots through an angle of 123°.
First, let's assume that the wiper blade sweeps out a circular arc with radius 24 inches (total arm and blade length). The length of this arc can be calculated using the formula for the length of a circular arc:
Arc Length = (θ/360°) * 2πr
where θ is the central angle in degrees and r is the radius.
In this case, the central angle is 123°, and the radius is 24 inches. Substituting these values into the formula, we get:
Arc Length = (123/360°) * 2π * 24
Now, we can use the formula for the area of a circular sector to calculate the area covered by this arc:
Area of Sector = (θ/360°) * πr^2
where θ is the central angle in degrees and r is the radius.
In our case, the central angle is 123°, and the radius is 24 inches. Substituting these values into the formula, we get:
Area of Sector = (123/360°) * π * 24^2
Finally, we subtract the area of the triangular portion (covered by the blade but not the arc) from the area of the circular sector to find the total area of the windshield being cleaned by the wiper:
Total Area = Area of Sector - Area of Triangle
To calculate the area of the triangle, we can use the formula for the area of a triangle:
Area of Triangle = (1/2) * base * height
In this case, the base of the triangle is the length of the blade (21 inches), and the height can be found using trigonometry. Since the blade pivots through an angle of 123°, the height can be calculated as the product of the length of the blade and the sine of half the angle:
Height = blade length * sin(θ/2)
Substituting the values into the formula, we get:
Area of Triangle = (1/2) * 21 * (21 * sin(123°/2))
Finally, we can substitute the values of the area of the sector and the area of the triangle into the equation to find the total area:
Total Area = (123/360°) * π * 24^2 - (1/2) * 21 * (21 * sin(123°/2))
By calculating the values on the right side of the equation, the result will give us the total area of the windshield being cleaned by the wiper.