From the top of a building 60 m high the angles of depression of the top and the bottom of a tower are observed to be $30^{\circ}$ and $60^{\circ}$. The height of the tower is:

Question

Accepted Answer

To determine the height of the tower, we can use trigonometry, specifically the tangent function, which relates the angles of depression to the heights and distances involved.

Let's denote:
- $ h $ as the height of the tower.
- $ d $ as the horizontal distance between the building and the tower.

From the top of the building, the angle of depression to the top of the tower is $ 30^\circ $, and to the bottom of the tower is $ 60^\circ $.

1. **Using the angle of depression to the top of the tower ($30^\circ$):**

The angle of depression is equal to the angle of elevation from the top of the tower to the top of the building. Therefore, we have:
   $$
   \tan(30^\circ) = \frac{60 - h}{d}
   $$
   $$
   \frac{1}{\sqrt{3}} = \frac{60 - h}{d}
   $$
   $$
   d = \sqrt{3}(60 - h)
   $$

2. **Using the angle of depression to the bottom of the tower ($60^\circ$):**

Similarly, the angle of depression is equal to the angle of elevation from the bottom of the tower to the top of the building. Therefore, we have:
   $$
   \tan(60^\circ) = \frac{60}{d}
   $$
   $$
   \sqrt{3} = \frac{60}{d}
   $$
   $$
   d = \frac{60}{\sqrt{3}}
   $$

3. **Equating the two expressions for $d$:**

From the two equations for $d$, we have:
   $$
   \sqrt{3}(60 - h) = \frac{60}{\sqrt{3}}
   $$

Solving for $h$:
   $$
   60 - h = \frac{60}{3}
   $$
   $$
   60 - h = 20
   $$
   $$
   h = 60 - 20 = 40
   $$

Thus, the height of the tower is 40 meters, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (80 m):** This would imply that the tower is taller than the building itself, which contradicts the given angles of depression. The calculations show that the height must be less than 60 m.

- **Option 3 (60 m):** This would mean the tower is as tall as the building, which is not possible given the angles of depression. The angle of depression to the top of the tower would be $0^\circ$ if they were the same height.

- **Option 4 (20 m):** This height does not satisfy the trigonometric relationships derived from the angles of depression. The calculations clearly show that the height must be 40 m.

Therefore, Option 1 is the correct answer, as it is consistent with the trigonometric analysis of the problem.