When seen from the top of a 90 meter high rock, the angles of depression of the top and bottom of a tower are 30° and 60°, respectively, then the height of that tower is :

Question

Accepted Answer

To determine the height of the tower, we can use trigonometric principles involving angles of depression and right triangles.

### Explanation:

1. **Understanding the Scenario:**
   - We have a rock that is 90 meters high.
   - From the top of this rock, the angles of depression to the top and bottom of a tower are 30° and 60°, respectively.

2. **Trigonometric Analysis:**
   - Let's denote:
     - $ h $ as the height of the tower.
     - $ d $ as the horizontal distance from the base of the rock to the base of the tower.

3. **Using the Angle of Depression to the Top of the Tower (30°):**
   - The angle of depression to the top of the tower is 30°, which means the angle of elevation from the top of the tower to the top of the rock is also 30°.
   - Using the tangent function: 
     $$
     \tan(30°) = \frac{\text{Height of the rock} - \text{Height of the tower}}{d} = \frac{90 - h}{d}
     $$
   - We know $\tan(30°) = \frac{1}{\sqrt{3}}$, so:
     $$
     \frac{1}{\sqrt{3}} = \frac{90 - h}{d}
     $$
   - This gives us: 
     $$
     d = \sqrt{3}(90 - h)
     $$

4. **Using the Angle of Depression to the Bottom of the Tower (60°):**
   - The angle of depression to the bottom of the tower is 60°, which means the angle of elevation from the base of the tower to the top of the rock is also 60°.
   - Using the tangent function:
     $$
     \tan(60°) = \frac{\text{Height of the rock}}{d} = \frac{90}{d}
     $$
   - We know $\tan(60°) = \sqrt{3}$, so:
     $$
     \sqrt{3} = \frac{90}{d}
     $$
   - This gives us:
     $$
     d = \frac{90}{\sqrt{3}}
     $$

5. **Equating the Two Expressions for $ d $:**
   - From the two equations for $ d $:
     $$
     \sqrt{3}(90 - h) = \frac{90}{\sqrt{3}}
     $$
   - Solving for $ h $:
     $$
     90 - h = \frac{90}{3}
     $$
     $$
     90 - h = 30
     $$
     $$
     h = 90 - 30 = 60
     $$

### Conclusion:

- The height of the tower is 60 meters, which corresponds to **Option 2**.

### Why Other Options are Incorrect:

- **Option 1 (45 meters):** This would imply incorrect calculations of the angles and distances, as shown by the trigonometric analysis.
- **Option 3 (75 meters):** This height would not satisfy the trigonometric relationships derived from the given angles of depression.
- **Option 4 (30 meters):** This would not align with the calculated height using the tangent functions for the given angles.

Thus, the correct answer is indeed **Option 2 (60 meters)**, as supported by the trigonometric calculations.