11. A car's rear windshield wiper rotates \( 125^{\circ} \). The total length of the wiper mechanism is 25 inche and the length of the wiper blade is 14 inches. Find the area wiped by the wiper blade. (Hint: You need to subtract two areas, to account for the part of the sector not wiped by the blade, near the base of the mechanism.)

To find the area wiped by the wiper blade, we first need to calculate the area of the entire sector swept by the mechanism. The formula for the area of a sector is: \[ \text{Area} = \frac{\theta}{360} \times \pi r^2 \] Here, \(\theta = 125^{\circ}\) and \(r = 25\) inches (the length of the wiper mechanism). \[ \text{Area of the sector} = \frac{125}{360} \times \pi (25)^2 \] Calculating this gives: \[ = \frac{125}{360} \times \pi \times 625 = \frac{125 \times 625 \times \pi}{360} \approx 273.89 \text{ square inches} \] Next, we calculate the area of the sector that remains unwiped at the base, where \(r = 11\) inches (the length of the wiper blade) since \(25 - 14 = 11\). \[ \text{Area of the unwiped sector} = \frac{125}{360} \times \pi (11)^2 \] Calculating this gives: \[ = \frac{125}{360} \times \pi \times 121 = \frac{125 \times 121 \times \pi}{360} \approx 42.08 \text{ square inches} \] Finally, subtract the unwiped sector area from the total swept area to find the area actually wiped by the wiper blade: \[ \text{Area wiped by the wiper blade} = 273.89 - 42.08 \approx 231.81 \text{ square inches} \] Thus, the area wiped by the wiper blade is approximately \(231.81\) square inches.