The angle of elevation of the top of a tower from a point on the ground is `60^(@)`. which is 25 m away from the foot of the tower. Find the height of the tower

Answer

To find the height of the tower given the angle of elevation and the distance from the tower, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem:**
We have a tower (let's denote it as AB) and a point on the ground (point C) that is 25 meters away from the foot of the tower (point B). The angle of elevation from point C to the top of the tower (point A) is 60 degrees.

2. **Draw a Diagram:**
Sketch a right triangle where:
- AB is the height of the tower (unknown).
- BC is the distance from the point on the ground to the foot of the tower, which is 25 meters.
- AC is the line of sight from point C to the top of the tower (point A).
- The angle ∠ACB is 60 degrees.

3. **Identify the Trigonometric Ratio:**
In this right triangle, we can use the tangent function because we have the opposite side (height of the tower AB) and the adjacent side (distance BC).
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
Here, \(\theta = 60^\circ\), Opposite = AB, and Adjacent = BC.

4. **Set Up the Equation:**
Using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BC} \]
We know that \(\tan(60^\circ) = \sqrt{3}\) and \(BC = 25\) meters. So we can write:
\[ \sqrt{3} = \frac{AB}{25} \]

5. **Solve for AB:**
Rearranging the equation to find AB:
\[ AB = 25 \cdot \sqrt{3} \]

6. **Calculate the Height:**
The height of the tower (AB) is:
\[ AB = 25\sqrt{3} \text{ meters} \]

### Final Result:
The height of the tower is \(25\sqrt{3}\) meters.

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