n'the figure below, à person standing at point A notices that the angle of elevation to the top of the antenna is 4 8° 30'. A second person standing 37.0 feet farther from the antenna chan the person at A finds the angle of elevation to the top of the antenna to be 41 15'. How far is the person al A from the base of the antenna?(Find X=?). 41° 15'
n'the figure below, à person standing at point A notices that the angle of elevation to the top of the antenna is 4 8° 30'. A second person standing 37.0 feet farther from the antenna chan the person at A finds the angle of elevation to the top of the antenna to be 41 15'. How far is the person al A from the base of the antenna?(Find X=?). 41° 15'
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Description:
A water tower is located 325 feet from a building. From a window in the building, it is observed that the angle of elevation to the top of the tower is 39° and the angle of depression to the bottom of the tower is 25°. How tall is the tower? How high is the window?
### Diagram Explanation:
The diagram shows a building and a water tower positioned 325 feet apart. Two angles are marked from the window in the building:
- **Angle of Elevation (39°):** This is the angle from the horizontal line (directly from the window) upwards to the top of the water tower.
- **Angle of Depression (25°):** This is the angle from the horizontal line downward to the bottom of the water tower.
The horizontal distance between the building and the water tower is clearly marked as 325 feet.
### Steps to Solve:
1. **Identify the right triangles:**
- For the angle of elevation, the triangle formed connects the window to the top of the tower.
- For the angle of depression, the triangle formed connects the window to the bottom of the tower.
2. **Use trigonometric functions:**
- Use the tangent function to find the height difference from the window to the top and bottom of the tower.
- For the top of the tower:
\[
\tan(39°) = \frac{\text{Height from window to top of tower}}{325}
\]
- For the bottom of the tower:
\[
\tan(25°) = \frac{\text{Height from window to bottom of tower}}{325}
\]
3. **Calculate:**
- Solve the equations to find:
- Height from the window to the top of the tower.
- Height from the window to the bottom of the tower.
4. **Determine the heights:**
- Subtract the height from the window to the bottom of the tower from the height from the window to the top to find the total height of the tower.
- The height from the ground to the window is the height calculated to the bottom of the tower.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07c38dc6-4a9f-400d-9784-ec1cee64c4c5%2Fcfdad82c-1cb9-438b-b6fd-2cd973e08fcf%2F1i4vhs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Description:
A water tower is located 325 feet from a building. From a window in the building, it is observed that the angle of elevation to the top of the tower is 39° and the angle of depression to the bottom of the tower is 25°. How tall is the tower? How high is the window?
### Diagram Explanation:
The diagram shows a building and a water tower positioned 325 feet apart. Two angles are marked from the window in the building:
- **Angle of Elevation (39°):** This is the angle from the horizontal line (directly from the window) upwards to the top of the water tower.
- **Angle of Depression (25°):** This is the angle from the horizontal line downward to the bottom of the water tower.
The horizontal distance between the building and the water tower is clearly marked as 325 feet.
### Steps to Solve:
1. **Identify the right triangles:**
- For the angle of elevation, the triangle formed connects the window to the top of the tower.
- For the angle of depression, the triangle formed connects the window to the bottom of the tower.
2. **Use trigonometric functions:**
- Use the tangent function to find the height difference from the window to the top and bottom of the tower.
- For the top of the tower:
\[
\tan(39°) = \frac{\text{Height from window to top of tower}}{325}
\]
- For the bottom of the tower:
\[
\tan(25°) = \frac{\text{Height from window to bottom of tower}}{325}
\]
3. **Calculate:**
- Solve the equations to find:
- Height from the window to the top of the tower.
- Height from the window to the bottom of the tower.
4. **Determine the heights:**
- Subtract the height from the window to the bottom of the tower from the height from the window to the top to find the total height of the tower.
- The height from the ground to the window is the height calculated to the bottom of the tower.
Transcribed Image Text:In the figure below, a person standing at point A notices that the angle of elevation to the top of the antenna is \(48^\circ 30'\). A second person standing 37.0 feet farther from the antenna than the person at A finds the angle of elevation to the top of the antenna to be \(41^\circ 15'\).
**Question:** How far is the person at A from the base of the antenna? (Find \(x\))
**Diagram Explanation:**
- The diagram shows two right triangles formed by the ground, the line of sight to the top of the antenna, and the vertical line from the top of the antenna to the ground.
- Point A corresponds to the location of the first person with an angle of elevation \(48^\circ 30'\).
- A vertical line of height \(h\) extends from the ground to the top of the antenna.
- Points are labeled:
- The person at A is a distance \(x\) from the base of the antenna.
- The second person is 37.0 feet farther from point A, making their distance from the antenna \(x + 37.0\) feet.
- The angle of elevation for the second person is \(41^\circ 15'\).
The problem requires finding the distance \(x\) between point A and the base of the antenna using the given angles and distances.
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