The arm and blade of a windshield wiper have a total length of 13 inches. If the blade is 9 inches long and the wiper sweeps out an angle of 99degrees​, how much window area can the blade​ clean?

To find the area of the window that the blade can clean, we need to calculate the area of a sector of a circle. The formula for the area of a sector is:

\[
\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2
\]

Where:
- \(\theta\) is the angle swept by the blade in degrees.
- \(r\) is the radius, which in this case is the length of the blade.

Given:
- The angle \(\theta = 99^\circ\).
- The length of the blade \(r = 9\) inches.

Now, plug these values into the formula:

\[
\text{Area} = \frac{99^\circ}{360^\circ} \times \pi \times (9 \, \text{inches})^2
\]

Let's calculate this step by step:

1. Calculate the fraction of the circle that the angle represents:

\[
\frac{99}{360} = 0.275
\]

2. Calculate the area of the full circle (with radius 9 inches):

\[
\pi \times 9^2 = \pi \times 81 \approx 254.47 \, \text{square inches}
\]

3. Multiply the fraction by the area of the full circle:

\[
0.275 \times 254.47 \approx 70 \, \text{square inches}
\]

Therefore, the windshield wiper blade can clean approximately **70 square inches** of the window.

Would you like more details or have any questions? Here are some related questions to expand on this topic:

1. What is the length of the wiper arm if the total length is 13 inches and the blade is 9 inches?
2. How would the cleaned area change if the angle swept by the wiper was increased to 120 degrees?
3. How would you calculate the area cleaned if the blade length was different?
4. What is the perimeter of the sector cleaned by the blade?
5. How would you calculate the arc length of the sector cleaned by the blade?

**Tip:** Always remember that the total length of a windshield wiper is the sum of both the arm and the blade, but only the blade length is considered when calculating the cleaning area.

Learn how to calculate the area cleaned by a windshield wiper blade sweeping an angle of 99 degrees. This problem involves geometry, specifically the area of a sector of a circle formula. Discover step-by-step solutions and related questions.

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