The angle of elevation of the top of a tower at a distance of 100 m from its foor on a horizontal plane is found to be `60^(@)`. Find the height of the tower.

Answer

To find the height of the tower using the angle of elevation, we can follow these steps:

### Step 1: Understand the Problem
We have a tower whose height we need to find. We know the angle of elevation from a point on the ground (100 meters away from the base of the tower) to the top of the tower is 60 degrees.

### Step 2: Draw a Diagram
Draw a right triangle where:
- The height of the tower is the vertical side (let's denote it as \( h \)).
- The distance from the base of the tower to the point of observation is the horizontal side (100 m).
- The angle of elevation is the angle between the horizontal line and the line of sight to the top of the tower (60 degrees).

### Step 3: Use Trigonometric Ratios
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side (height of the tower) to the adjacent side (distance from the tower). Therefore, we can write:

\[ \tan(60^\circ) = \frac{h}{100} \]

### Step 4: Substitute the Value of \( \tan(60^\circ) \)
From trigonometric tables, we know that:

\[ \tan(60^\circ) = \sqrt{3} \approx 1.732 \]

So we can substitute this value into our equation:

\[ \sqrt{3} = \frac{h}{100} \]

### Step 5: Solve for \( h \)
To find \( h \), multiply both sides of the equation by 100:

\[ h = 100 \cdot \sqrt{3} \]

### Step 6: Calculate the Height
Now, we can calculate the height using the approximate value of \( \sqrt{3} \):

\[ h \approx 100 \cdot 1.732 = 173.2 \text{ meters} \]

### Final Answer
The height of the tower is approximately \( 173.2 \) meters.

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