Question

Windshield Wipers The top and bottom ends of a windshield wiper blade are 34 in. and 14 in., respectively, from the pivot point. While in operation, the wiper sweeps through$\mathbf{135}\mathbf{°}$. Find the area swept by the blade.

Short answer

The area swept by the blade is $\mathbf{1131}\mathit{i}{\mathit{n}}^{\mathbf{2}}$.

Step-by-step solution

Step 1. Given information.

The areaA of a sector with central angle of $\theta$radians is:

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

The area swept is the difference of the larger $\left(r_2=34\right.in)$in and smaller $\left(r_1=14\right.in)$sectors that is:

$A=\frac{1}{2}\left(r_2^2-r_1^2\right)\times \theta$

To convert $\theta$into radians, multiply the angle by $\frac{\pi }{180}$.

Step 2. Finding the area swept by the blade.

To convert $\theta$into radians multiply with $\frac{\pi }{180}$:

$\theta ={135}^{°}\times \frac{\pi }{{180}^{°}}\text{rad}$

$=\frac{3\pi }{4}\text{rad}$

The area swept by the blade is:

$A=\frac{1}{2}\left(r_2^2-r_1^2\right)\times \theta$

Put given values in the above formula, we get:

$=\frac{1}{2}\left({34}^2-{14}^2\right)\times \frac{3\pi }{4}$

$=\frac{1}{2}(1156-196)\times \frac{3\times 3.14}{4}(\because \pi =3.14)$

localid="1648731327876" role="math" $=\frac{960\times 3\times 3.14}{2\times 4}$

$=\frac{9043.2}{8}$

$\approx 1130.97335529$

$\approx 1131{\text{in}}^2$