The angle of depression of a point situated at a distance of 70 m from the base of a tower is `60^(@)`.The height of the tower is :

Options

A. `25sqrt(3)m`
B. `70sqrt(3)m`
C. `(70sqrt(3))/(3)`m
D. 70 m

Answer

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

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a tower and a point on the ground that is 70 meters away from the base of the tower. The angle of depression from the top of the tower to this point is 60 degrees.

2. **Draw a Diagram**:
Draw a right triangle where:
- The height of the tower is represented as \( AB \).
- The distance from the base of the tower to the point on the ground is \( BC = 70 \) meters.
- The angle of depression \( \angle ACB = 60^\circ \).

3. **Identify the Right Triangle**:
In triangle \( ABC \), we know:
- \( AC \) is the line of sight from the top of the tower to the point on the ground.
- \( AB \) is the height of the tower (which we need to find).
- \( BC \) is the horizontal distance from the base of the tower to the point (70 meters).

4. **Use Trigonometric Ratios**:
The angle of depression from point A to point C is equal to the angle of elevation from point C to point A. Therefore, we can use the tangent function:
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Here, \( AB \) is the height of the tower and \( BC = 70 \) meters.

5. **Calculate \( \tan(60^\circ) \)**:
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]

6. **Set Up the Equation**:
Substitute the values into the equation:
\[ \sqrt{3} = \frac{AB}{70} \]

7. **Solve for \( AB \)**:
Rearranging gives:
\[ AB = 70 \times \sqrt{3} \]

8. **Calculate the Height**:
The height of the tower is:
\[ AB = 70\sqrt{3} \text{ meters} \]

### Final Answer:
The height of the tower is \( 70\sqrt{3} \) meters.

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