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-angled triangles. Here’s a step-by-step solution:

### Step 1: Understand the scenario
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 given as \(45^\circ\).

### Step 2: Draw a diagram
Draw a vertical line representing the tower. Label the top of the tower as point A and the base as point B. Draw a horizontal line from point A to the point on the ground (point C) that is 70 meters away from point B. The angle of depression from point A to point C is \(45^\circ\).

### Step 3: Identify the right triangle
In the right triangle ABC:
- AB is the height of the tower (which we need to find).
- BC is the distance from the base of the tower to the point on the ground, which is 70 meters.
- Angle ACB is \(45^\circ\).

### Step 4: Use the tangent function
In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Therefore, we can write:
\[ \tan(\angle ACB) = \frac{AB}{BC} \]
Substituting the known values:
\[ \tan(45^\circ) = \frac{AB}{70} \]

### Step 5: Calculate the height of the tower
We know that \(\tan(45^\circ) = 1\). Therefore, we can set up the equation:
\[ 1 = \frac{AB}{70} \]
This implies:
\[ AB = 70 \times 1 = 70 \text{ meters} \]

### Conclusion
The height of the tower is \(70\) meters.

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