Questions: A car's rear windshield wiper rotates 135°. The total length of the wiper mechanism is 24 inches and the length of the wiper blade is 13 inches. Find the area wiped by the wiper blade. (Round your answer to one decimal place.)

Solution Steps

To find the area wiped by the wiper blade, we can model the problem as a sector of a circle. The wiper blade sweeps out a sector with a central angle of \(135^{\circ}\). The radius of this sector is the length of the wiper blade, which is 13 inches. The area of a sector of a circle is given by the formula:

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

where \(\theta\) is the central angle in degrees and \(r\) is the radius.

We are given the following parameters:

• Central angle \( \theta = 135^{\circ} \)
• Radius of the sector (length of the wiper blade) \( r = 13 \) inches.

The area \( A \) of a sector of a circle is calculated using the formula: \[ A = \frac{\theta}{360} \times \pi \times r^2 \] Substituting the known values: \[ A = \frac{135}{360} \times \pi \times (13)^2 \]

Calculating the area: \[ A = \frac{135}{360} \times \pi \times 169 \] This simplifies to: \[ A \approx 199.09843442125313 \] Rounding to one decimal place gives: \[ A \approx 199.1 \]

Final Answer