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.)

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We model the area wiped by the wiper as the difference between two sectors. The full wiper mechanism has a length (radius) of 25 inches, but the wiper blade is only 14 inches long. Thus, the region actually wiped is the area of the sector of a circle with radius 25 inches minus the area of the sector (near the base) that is not reached by the blade. This inner sector has a radius of  
$$
25 - 14 = 11 \text{ inches}.
$$

Both sectors have the same central angle of $$ 125^\circ $$.

1. The area of a sector of a circle with radius $$ r $$ and central angle $$ \theta $$ (in degrees) is given by  
   $$
   \text{Area} = \frac{\theta}{360^\circ} \pi r^2.
   $$

2. For the outer sector (radius 25 inches) the area is  
   $$
   A_{\text{outer}} = \frac{125}{360} \pi (25)^2 = \frac{125}{360} \pi (625).
   $$

3. For the inner sector (radius 11 inches) the area is  
   $$
   A_{\text{inner}} = \frac{125}{360} \pi (11)^2 = \frac{125}{360} \pi (121).
   $$

4. The area wiped by the blade is the difference between these two sectors:  
   $$
   A = A_{\text{outer}} - A_{\text{inner}} = \frac{125}{360} \pi \left(625 - 121\right).
   $$
   Simplify the expression inside the parentheses:
   $$
   625 - 121 = 504.
   $$
   Thus,
   $$
   A = \frac{125}{360} \pi \cdot 504.
   $$

5. To simplify, note that
   $$
   \frac{504}{360} = \frac{504 \div 72}{360 \div 72} = \frac{7}{5}.
   $$
   Therefore,
   $$
   A = \frac{125}{1} \pi \cdot \frac{7}{5} = \frac{125 \times 7}{5} \pi.
   $$
   Since
   $$
   \frac{125}{5} = 25,
   $$
   we obtain:
   $$
   A = 25 \times 7 \pi = 175 \pi.
   $$

Thus, the area wiped by the wiper blade is  
$$
\boxed{175\pi \text{ square inches}}.
$$
The area wiped by the wiper blade is $$ 175\pi $$ square inches.

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