Certainly! Let's break down the solution step-by-step:
1. Identify the Given Data:
- Distance from the man to the tower: [tex]\(60\)[/tex] meters
- Angle of elevation to the top of the flagpole: [tex]\(62^\circ\)[/tex]
- Angle of elevation to the bottom of the flagpole: [tex]\(62^\circ\)[/tex]
2. Understanding the Angles:
Both the angles of elevation (to the top and bottom of the flagpole) are given as [tex]\(62^\circ\)[/tex]. This implies that, essentially, the top and the bottom of the flagpole are seen from the same angle of elevation.
3. Using Trigonometry to Find Heights:
We can use the tangent function ([tex]\(\tan\)[/tex]), which relates the angle of elevation to the ratio of the opposite side (height of the object) to the adjacent side (distance from the observer):
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
4. Calculate the Height to the Top of the Flagpole:
First, we convert the angle from degrees to radians because the tangent function usually works with radians in some contexts.
[tex]\[
\theta = 62^\circ = \text{Convert to radians}
\][/tex]
[tex]\[
\text{height to top} = 60 \times \tan(62^\circ)
\][/tex]
5. Similarly, Calculate the Height to the Bottom of the Flagpole:
Again, using the same angle of elevation:
[tex]\[
\text{height to bottom} = 60 \times \tan(62^\circ)
\][/tex]
6. Calculate the Height of the Flagpole:
The height of the flagpole is the difference between the height to the top of the flagpole and the height to the bottom of the flagpole.
[tex]\[
\text{height of flagpole} = \text{height to top} - \text{height to bottom}
\][/tex]
Given that both angles of elevation are the same, the calculations result in:
[tex]\[
\text{height to top} = 112.8 \, m \quad \text{(approximately)}
\][/tex]
[tex]\[
\text{height to bottom} = 112.8 \, m \quad \text{(approximately)}
\][/tex]
Thus:
[tex]\[
\text{height of flagpole} = 112.8 \, m - 112.8 \, m = 0.0 \, m
\][/tex]
Therefore, the height of the flagpole is:
~0.0 meters.
