Question

The angle of elevation of the top of the flag post from a point A on a level ground is
13











The angle of elevation of the top of the flag post from a point A on a level ground is 13 degrees.The angle of elevation of the top of the flag post from another point B nearer the flag post and 120 metres is 30 degrees.B is between A and the bottom of the flag post and the three points are collinear.find the distance from point B to the top of the flag post.




T

Answers

Answered by GPT-4o mini
To solve the problem, we need to set up the situation using basic trigonometry. Let's define some variables and work step by step.

1. **Define the variables:**
- Let \( h \) = height of the flag pole.
- Let \( d \) = distance from point A to the base of the flag pole.
- We know the angle of elevation from point A is \( 13^\circ \) and from point B it is \( 30^\circ \).
- The distance between point A and point B is \( 120 \) meters.

Therefore, we can write:
\[
d - 120 = \text{distance from point A to point B}
\]

2. **Using the angle from point A:**
From point A:
\[
\tan(13^\circ) = \frac{h}{d}
\]
Rearranging gives us:
\[
h = d \cdot \tan(13^\circ)
\]

3. **Using the angle from point B:**
From point B, whose distance from the flag pole is \( d - 120 \):
\[
\tan(30^\circ) = \frac{h}{d - 120}
\]
Rearranging gives us:
\[
h = (d - 120) \cdot \tan(30^\circ)
\]

4. **Setting the equations equal:**
We have two expressions for \( h \):
\[
d \cdot \tan(13^\circ) = (d - 120) \cdot \tan(30^\circ)
\]

5. **Solving for \( d \):**
Expanding and rearranging:
\[
d \cdot \tan(13^\circ) = d \cdot \tan(30^\circ) - 120 \cdot \tan(30^\circ
\]
\[
d \cdot \tan(13^\circ) - d \cdot \tan(30^\circ) = -120 \cdot \tan(30^\circ)
\]
Factoring out \( d \):
\[
d (\tan(13^\circ) - \tan(30^\circ)) = -120 \cdot \tan(30^\circ)
\]
Therefore:
\[
d = \frac{-120 \cdot \tan(30^\circ)}{\tan(13^\circ) - \tan(30^\circ)}
\]

6. **Calculating \( d \):**
Using \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.5774 \) and \( \tan(13^\circ) \approx 0.2309 \):
\[
d = \frac{-120 \cdot \frac{1}{\sqrt{3}}}{0.2309 - 0.5774}
\]
\[
d = \frac{-120 \cdot \frac{1}{\sqrt{3}}}{-0.3465} \approx \frac{120 \cdot 0.5774}{0.3465} \approx \frac{69.288}{0.3465} \approx 200.0
\]

7. **Finding \( h \) using \( d \):**
Now substitute \( d \) back into \( h = d \cdot \tan(13^\circ) \):
\[
h \approx 200.0 \cdot 0.2309 \approx 46.18
\]

8. **Finding the distance from point B to the top of the flag post:**
From point B, the height remains 46.18 meters, and the base distance from B to the pole is:
\[
d - 120 = 200 - 120 = 80 \text{ meters}
\]
Now we can find the distance from point B to the top of the flag post using the Pythagorean theorem:
\[
\text{Distance from B to top} = \sqrt{h^2 + (d - 120)^2} = \sqrt{(46.18)^2 + (80)^2}
\]
\[
= \sqrt{2135.056 + 6400} \approx \sqrt{8535.056} \approx 92.58 \text{ meters}
\]

Thus, the distance from point B to the top of the flag post is approximately **92.58 meters**.
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