The angle of depression of a point situated at a distance of 70 metres from the base of a tower is `45^@`. The height of the tower is

Options

A. 70 m
B. `70sqrt2` m
C. `70/sqrt2` m
D. 35 m

Answer

To solve the problem, we will use the concept of angles of depression and right triangles.

### Step-by-Step Solution

1. **Understand the Problem**: We have a tower (let's denote it as AB) and a point C that is 70 meters away from the base of the tower (point B). The angle of depression from the top of the tower (point A) to the point C is given as 45 degrees.

2. **Draw a Diagram**: Visualize the scenario. Draw a vertical line representing the tower (AB), where A is the top of the tower and B is the base. Point C is located horizontally 70 meters away from B. The angle of depression from A to C is 45 degrees.

3. **Identify Angles**: The angle of depression from A to C is 45 degrees. By the properties of alternate interior angles, the angle of elevation from C to A is also 45 degrees.

4. **Set Up the Right Triangle**: In triangle ABC, where:
- AB is the height of the tower (h),
- BC is the distance from the base of the tower to point C (70 meters),
- Angle ACB is 45 degrees.

5. **Use Trigonometric Ratios**: We can use the tangent function, which relates the opposite side (height of the tower) to the adjacent side (distance from the tower):
\[ \tan(45^\circ) = \frac{AB}{BC} \]
Here, \( AB = h \) and \( BC = 70 \).

6. **Calculate Using the Tangent Value**: Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h}{70} \]

7. **Cross Multiply to Solve for h**:
\[ h = 70 \times 1 = 70 \text{ meters} \]

8. **Conclusion**: The height of the tower is 70 meters.

### Final Answer
The height of the tower is **70 meters**.