Question

An automobile windshield wiper 7 inches long rotates through an angle of 60 ∘ . If the rubber part of the blade covers only the last 6 inches of the wiper, find the area of the windshield cleaned by the windshield wiper.

Answer 1

Answer:

25.13 square inch

Step-by-step explanation:

The automobile windshield wiper 7 inches long rotates through an angle of 60 degrees.

Note that the rubber part covers only the last 6 inches of the wiper.

To find the area of the windshield cleaned by the wiper,

We find the area of the annulus shaded in grey in the diagram.

Area of a Sector=$\frac{\theta}{360}X\pi r^2$

For the larger sector, Radius, R=7cm

For the smaller Sector, radius r=1cm

Area cleaned by the wiper

$=\frac{\theta}{360}X\pi R^2-\frac{\theta}{360}X\pi r^2\\=\frac{\theta}{360}X\pi(R^2-r^2)\\=\frac{60}{360}X\pi(49-1)\\=\frac{48 X \pi}{6}\\=25.13$

The area cleaned by the wiper is 25.13 square inch.

Answer 2

Thus the required area is $0.096$ square inches.

Area of the windshield:

Acute area of the windshield glazing means the rectangular area of the windshield, eight and one-half inches by 11 inches, directly in front of the driver's line of vision as depicted.

Let the area of the windshield ABC be $A$ then,

$A=\frac{1}{2}r^2\theta$

Now, substituting the given values we get,

$A=\frac{1}{2}r^2\theta\\A=\frac{1}{2}\times(7)^2 \times \frac{\pi}{180} \\A=0.4276$

Let the area of the windshield ADE be $A'$ then,

$A'=\frac{1}{2}r^2\theta$

Now, substituting the given values we get,

$A'=\frac{1}{2}r^2\theta\\A'=\frac{1}{2}\times(1)^2 \times 60\times \frac{\pi}{180} \\A'=0.5236$

For finding the area of the windshield takes the difference of $A\ and\ A'$ then,

$0.5236-0.4276=0.096$

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