Answer :
Certainly! Let's solve this problem step-by-step.
### Given:
1. The angles of depression from the observation tower to points A and B are [tex]\(7^\circ\)[/tex] and [tex]\(13^\circ\)[/tex] respectively.
2. The distance between points A and B is 1 kilometer.
3. The points and the observation tower are in the same vertical plane.
### Definitions:
1. Angle of depression is the angle between the horizontal line from the observer and the line of sight to the observed point.
2. With these angles of depression, we can calculate the angles of elevation from points A and B to the top of the tower, which were [tex]\(90^\circ\)[/tex] minus the angles of depression.
### Steps to solve the problem:
#### 1. Calculate the angles of elevation:
- Angle of elevation from point A ([tex]\( \theta_A \)[/tex]):
[tex]\[ \theta_A = 90^\circ - 7^\circ = 83^\circ \][/tex]
- Angle of elevation from point B ([tex]\( \theta_B \)[/tex]):
[tex]\[ \theta_B = 90^\circ - 13^\circ = 77^\circ \][/tex]
#### 2. Use trigonometry to find the height of the tower relative to point B:
Since point B is closer along the horizontal than point A, we first find the height from point B, which we represent as [tex]\( h_B \)[/tex].
- Using the tangent function which relates the height ([tex]\(h\)[/tex]) and the horizontal distance ([tex]\(d\)[/tex]) through the angle of elevation ([tex]\( \theta \)[/tex]):
[tex]\[ \tan(\theta_B) = \frac{h_B}{1 \text{ km}} \][/tex]
[tex]\[ h_B = 1 \text{ km} \times \tan(77^\circ) \][/tex]
The result for [tex]\( h_B \)[/tex] is:
[tex]\[ h_B \approx 0.23087 \text{ km} \][/tex]
This value was calculated using the tangent of [tex]\(77^\circ\)[/tex], giving us the height tower relative to point B.
#### 3. Determine the total height of the tower from point A:
To determine the full height of the observation tower, we need to account for the difference in elevation angles. Since point B is 1 km away horizontally from point A:
[tex]\[ h_{\text{total}} = 1 \text{ km} \times (\tan(77^\circ) - \tan(83^\circ)) \][/tex]
The height difference between these two angles of elevation gives the additional height from point A.
The calculation yields:
[tex]\[ h_{\text{total}} \approx 0.10808 \text{ km} \][/tex]
So, the total height of the observation tower is approximately [tex]\( 0.10808 \)[/tex] km or 108.08 meters.
#### Final Result:
- Height from point B ([tex]\( h_B \)[/tex]): [tex]\( 0.23087 \)[/tex] km (230.87 meters)
- Total height ([tex]\( h_{\text{total}} \)[/tex]): [tex]\( 0.10808 \)[/tex] km (108.08 meters)
The height of the observation tower is 108.08 meters from point A as calculated.
### Given:
1. The angles of depression from the observation tower to points A and B are [tex]\(7^\circ\)[/tex] and [tex]\(13^\circ\)[/tex] respectively.
2. The distance between points A and B is 1 kilometer.
3. The points and the observation tower are in the same vertical plane.
### Definitions:
1. Angle of depression is the angle between the horizontal line from the observer and the line of sight to the observed point.
2. With these angles of depression, we can calculate the angles of elevation from points A and B to the top of the tower, which were [tex]\(90^\circ\)[/tex] minus the angles of depression.
### Steps to solve the problem:
#### 1. Calculate the angles of elevation:
- Angle of elevation from point A ([tex]\( \theta_A \)[/tex]):
[tex]\[ \theta_A = 90^\circ - 7^\circ = 83^\circ \][/tex]
- Angle of elevation from point B ([tex]\( \theta_B \)[/tex]):
[tex]\[ \theta_B = 90^\circ - 13^\circ = 77^\circ \][/tex]
#### 2. Use trigonometry to find the height of the tower relative to point B:
Since point B is closer along the horizontal than point A, we first find the height from point B, which we represent as [tex]\( h_B \)[/tex].
- Using the tangent function which relates the height ([tex]\(h\)[/tex]) and the horizontal distance ([tex]\(d\)[/tex]) through the angle of elevation ([tex]\( \theta \)[/tex]):
[tex]\[ \tan(\theta_B) = \frac{h_B}{1 \text{ km}} \][/tex]
[tex]\[ h_B = 1 \text{ km} \times \tan(77^\circ) \][/tex]
The result for [tex]\( h_B \)[/tex] is:
[tex]\[ h_B \approx 0.23087 \text{ km} \][/tex]
This value was calculated using the tangent of [tex]\(77^\circ\)[/tex], giving us the height tower relative to point B.
#### 3. Determine the total height of the tower from point A:
To determine the full height of the observation tower, we need to account for the difference in elevation angles. Since point B is 1 km away horizontally from point A:
[tex]\[ h_{\text{total}} = 1 \text{ km} \times (\tan(77^\circ) - \tan(83^\circ)) \][/tex]
The height difference between these two angles of elevation gives the additional height from point A.
The calculation yields:
[tex]\[ h_{\text{total}} \approx 0.10808 \text{ km} \][/tex]
So, the total height of the observation tower is approximately [tex]\( 0.10808 \)[/tex] km or 108.08 meters.
#### Final Result:
- Height from point B ([tex]\( h_B \)[/tex]): [tex]\( 0.23087 \)[/tex] km (230.87 meters)
- Total height ([tex]\( h_{\text{total}} \)[/tex]): [tex]\( 0.10808 \)[/tex] km (108.08 meters)
The height of the observation tower is 108.08 meters from point A as calculated.
