Chapter 4.1, Problem 70E

To determine

To find: the area wiped by the wiper blade.

Answer to Problem 70E

  $A\approx 549.779{\text{ in}}^2$

Explanation of Solution

Given:

A car’s rear windshield wiper rotates $125°$ . The total length of the wiper mechanism is 25 inches and the length of the wiper blade is 14 inches.

Concept Used:

Area of sector of circle:

For a circle of radius r , the area A of a sector of the circle with central angle $\theta$ (in radian measure) is:

  $A=\frac{1}{2}{r}^2\theta$

First convert the central angle from degree measure to radian measure.

  $\begin{array}{c}\theta =125°\\ =125°\times \frac{\pi }{180°}\\ =\frac{25\pi }{36}\end{array}$

The total length of the wiper mechanism is 25 inches.

  $r_1=25\text{ in}$

Thus, the area covered by the whole wiper mechanism is,

  $\begin{array}{c}{A}_1=\frac{1}{2}{r}_1{}^2\theta \\ =\frac{1}{2}\times {\left(25\right)}^2\times \left(\frac{25\pi }{36}\right)\\ \approx 681.769{\text{ in}}^2\end{array}$

The length of the wiper blade is 14 inches.

  $r_2=14\text{ in}$

Thus the length that is not wiping is:

  $\begin{array}{c}r=25-14\\ =11\text{ in}\end{array}$

Thus, the area that is not being wiped is,

  $\begin{array}{c}{A}_2=\frac{1}{2}{r}^2\theta \\ =\frac{1}{2}\times {\left(11\right)}^2\times \left(\frac{25\pi }{36}\right)\\ \approx 131.990{\text{ in}}^2\end{array}$

Then, the area being wiped is,

  $\begin{array}{c}A=A_1-A_2\\ \approx 681.769-131.990\\ \approx 549.779{\text{ in}}^2\end{array}$