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

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Draw a Diagram**:
- Draw a right triangle where:
- Point A is the top of the tower.
- Point B is the foot of the tower.
- Point C is the point on the ground 100 meters away from the foot of the tower.
- The angle of elevation from point C to point A is \(60^\circ\).

2. **Label the Triangle**:
- Let the height of the tower (AB) be \(h\) meters.
- The distance from point C to the foot of the tower (BC) is 100 meters.

3. **Use the Tangent Function**:
- In triangle ABC, we can use the tangent of the angle of elevation:
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
- Here, \(AB = h\) and \(BC = 100\) meters.

4. **Substitute the Values**:
- Substitute the known values into the equation:
\[ \tan(60^\circ) = \frac{h}{100} \]
- We know that \(\tan(60^\circ) = \sqrt{3}\).

5. **Set Up the Equation**:
- Now, the equation becomes:
\[ \sqrt{3} = \frac{h}{100} \]

6. **Solve for \(h\)**:
- Cross-multiply to solve for \(h\):
\[ h = 100 \cdot \sqrt{3} \]

7. **Calculate the Height**:
- If we want to find the numerical value, we can use the approximate value of \(\sqrt{3} \approx 1.732\):
\[ h \approx 100 \cdot 1.732 = 173.2 \text{ meters} \]

### Final Answer:
The height of the tower is \(100\sqrt{3}\) meters or approximately \(173.2\) meters.

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