The total arm and blade of a windshield wiper is 11 in. long and rotates back and forth through an angle of 80°. The shaded region in the figure is the portion of the windshield cleaned by the 8-in. wiper blade. What is the area of the region cleaned? 80° 11 in. 8 in.
The total arm and blade of a windshield wiper is 11 in. long and rotates back and forth through an angle of 80°. The shaded region in the figure is the portion of the windshield cleaned by the 8-in. wiper blade. What is the area of the region cleaned? 80° 11 in. 8 in.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Windshield Wiper Blade Sweep Calculation**
The total arm and blade of a windshield wiper is 11 inches long and rotates back and forth through an angle of 80°. The shaded region in the figure is the portion of the windshield cleaned by the 8-inch wiper blade. What is the area of the region cleaned?
### Diagram Explanation:
- The diagram illustrates a sector of a circle with a central angle of 80°.
- The total wiper arm length is 11 inches, and the length of the wiper blade contributing to the swept area is 8 inches.
- The shaded area represents the difference between the sweep of the entire wiper arm and the portion of the windshield that is not reached by the 8-inch blade.
### Calculation of Cleaned Area:
1. Calculate the area swept by the entire arm and blade (radius = 11 inches).
2. Subtract the area not cleaned by the part of the blade beyond the 8-inch mark (radius = 8 inches).
#### Step-by-Step Solution:
1. **Area of Circle Formula**:
\[ \text{Area} = \pi r^2 \]
2. **Area of Sector Formula**:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Where \( \theta \) is the central angle.
3. **Area of the region cleaned by the 11-inch arm**:
\[ \text{Area}_{11} = \frac{80^\circ}{360^\circ} \times \pi \times 11^2 \]
4. **Area of the region not cleaned by the 8-inch blade**:
\[ \text{Area}_{8} = \frac{80^\circ}{360^\circ} \times \pi \times 8^2 \]
5. **Effective Cleaned Area**:
\[ \text{Effective Cleaned Area} = \text{Area}_{11} - \text{Area}_{8} \]
\[ \text{Area}_{11} = \frac{80}{360} \times \pi \times 121 \]
\[ \text{Area}_{11} = \frac{80}{360} \times \pi \times 64 \]
Finally, subtract the smaller area from the larger to find the cleaned area.
The area of the region cleaned is \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2bbef39-13a9-4a3a-ae0f-bc83a28a95a1%2F56c3a51e-a7b6-4310-bc5c-b414a1ecb6ad%2Fwkrl6nr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Windshield Wiper Blade Sweep Calculation**
The total arm and blade of a windshield wiper is 11 inches long and rotates back and forth through an angle of 80°. The shaded region in the figure is the portion of the windshield cleaned by the 8-inch wiper blade. What is the area of the region cleaned?
### Diagram Explanation:
- The diagram illustrates a sector of a circle with a central angle of 80°.
- The total wiper arm length is 11 inches, and the length of the wiper blade contributing to the swept area is 8 inches.
- The shaded area represents the difference between the sweep of the entire wiper arm and the portion of the windshield that is not reached by the 8-inch blade.
### Calculation of Cleaned Area:
1. Calculate the area swept by the entire arm and blade (radius = 11 inches).
2. Subtract the area not cleaned by the part of the blade beyond the 8-inch mark (radius = 8 inches).
#### Step-by-Step Solution:
1. **Area of Circle Formula**:
\[ \text{Area} = \pi r^2 \]
2. **Area of Sector Formula**:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Where \( \theta \) is the central angle.
3. **Area of the region cleaned by the 11-inch arm**:
\[ \text{Area}_{11} = \frac{80^\circ}{360^\circ} \times \pi \times 11^2 \]
4. **Area of the region not cleaned by the 8-inch blade**:
\[ \text{Area}_{8} = \frac{80^\circ}{360^\circ} \times \pi \times 8^2 \]
5. **Effective Cleaned Area**:
\[ \text{Effective Cleaned Area} = \text{Area}_{11} - \text{Area}_{8} \]
\[ \text{Area}_{11} = \frac{80}{360} \times \pi \times 121 \]
\[ \text{Area}_{11} = \frac{80}{360} \times \pi \times 64 \]
Finally, subtract the smaller area from the larger to find the cleaned area.
The area of the region cleaned is \(
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