If a sprinkler 7 meters from one end of a 4- meter long wall has coverage to the end of the other end of the wall, what is the angle of the arc the sprinkler turns through? Refer to the image of a poorly drawn sprinkler below. Use exact values involving trigonometric functions, and also a decimal approximation, in degrees.
If a sprinkler 7 meters from one end of a 4- meter long wall has coverage to the end of the other end of the wall, what is the angle of the arc the sprinkler turns through? Refer to the image of a poorly drawn sprinkler below. Use exact values involving trigonometric functions, and also a decimal approximation, in degrees.
Algebra and Trigonometry (6th Edition)
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![**Problem 3: Calculating the Sprinkler's Arc Angle**
A sprinkler, located 7 meters from one end of a 4-meter long wall, is able to direct its water to the end of the wall. The task is to determine the angle (in degrees) of the arc through which the sprinkler rotates.
**Diagram Explanation:**
The diagram depicts a section of a 4-meter wall meeting a fence perpendicularly. The sprinkler is positioned 7 meters away from one end of the wall, along the fence line. Curved lines show the paths of the sprinkler’s water arc reaching the end of the wall.
**Solution Steps:**
1. **Understand the Triangle:**
- The scenario can be framed as a right triangle where:
- The horizontal distance along the fence from the sprinkler to the end of the wall is 7 meters.
- The wall distance (perpendicular to the fence) is 4 meters.
2. **Calculate with Trigonometry:**
- Use the tangent function, which relates the opposite side to the adjacent side of a right triangle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{7}
\]
- Solve for \(\theta\):
\[
\theta = \tan^{-1}\left(\frac{4}{7}\right)
\]
3. **Convert to Degrees:**
- Calculate \(\theta\) to find the angle in degrees using a calculator or trigonometric tables.
- Approximate to get a decimal value as needed.
This results in the angle \(\theta\), representing the sprinkler's angle of coverage in degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9072c5f8-5632-4fa6-afcd-5fe35f88b36f%2Faa00cb6c-5cba-4178-9815-284704fa5414%2Frgrjmoj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Calculating the Sprinkler's Arc Angle**
A sprinkler, located 7 meters from one end of a 4-meter long wall, is able to direct its water to the end of the wall. The task is to determine the angle (in degrees) of the arc through which the sprinkler rotates.
**Diagram Explanation:**
The diagram depicts a section of a 4-meter wall meeting a fence perpendicularly. The sprinkler is positioned 7 meters away from one end of the wall, along the fence line. Curved lines show the paths of the sprinkler’s water arc reaching the end of the wall.
**Solution Steps:**
1. **Understand the Triangle:**
- The scenario can be framed as a right triangle where:
- The horizontal distance along the fence from the sprinkler to the end of the wall is 7 meters.
- The wall distance (perpendicular to the fence) is 4 meters.
2. **Calculate with Trigonometry:**
- Use the tangent function, which relates the opposite side to the adjacent side of a right triangle:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{7}
\]
- Solve for \(\theta\):
\[
\theta = \tan^{-1}\left(\frac{4}{7}\right)
\]
3. **Convert to Degrees:**
- Calculate \(\theta\) to find the angle in degrees using a calculator or trigonometric tables.
- Approximate to get a decimal value as needed.
This results in the angle \(\theta\), representing the sprinkler's angle of coverage in degrees.
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