Question

If a windshield wiper covers an area of approximately 190 square inches when it rotates at an angle of 17π30
radians, find the length of the wiper to the nearest tenth of an inch.

Answer 1

To find the length of the windshield wiper (which acts as the radius of the circular arc it sweeps), we can use the formula for the area \( A \) of a sector of a circle:

\[
A = \frac{1}{2} r^2 \theta
\]

where:
- \( A \) is the area of the sector,
- \( r \) is the radius (length of the wiper), and
- \( \theta \) is the angle in radians.

Given:
- \( A = 190 \) square inches
- \( \theta = \frac{17\pi}{30} \) radians

We can rearrange the formula to solve for \( r \):

\[
r^2 = \frac{2A}{\theta}
\]

Now, substituting the given values into the formula:

\[
r^2 = \frac{2 \times 190}{\frac{17\pi}{30}}
\]

Calculating the numerator:

\[
2 \times 190 = 380
\]

Now substitute this into the equation:

\[
r^2 = \frac{380}{\frac{17\pi}{30}} = 380 \times \frac{30}{17\pi}
\]

Next, simplifying further:

\[
r^2 = \frac{11400}{17\pi}
\]

Now we will calculate this value. Using \( \pi \approx 3.14159 \):

\[
r^2 = \frac{11400}{17 \times 3.14159} \approx \frac{11400}{53.40703} \approx 213.144
\]

Now, we take the square root to find \( r \):

\[
r \approx \sqrt{213.144} \approx 14.6
\]

Thus, the length of the wiper to the nearest tenth of an inch is:

\[
\boxed{14.6} \text{ inches}
\]