The angle of elevation of the top of a vertical tower situated perpendicularly on a plane is observed as `60^(@)` from a point P on the same plane .From another point Q , 10 m vertically above the point P , the angle of depression of the foot of the tower is `30^(@)` The height of the tower is

Options

A. 15 m
B. 30 m
C. 20 m
D. 25 m

Answer

To solve the problem, we need to find the height of the tower based on the given angles of elevation and depression. Let's break it down step by step.

### Step 1: Understand the Setup
We have a vertical tower, and we are given two points:
- Point P on the ground, from where the angle of elevation to the top of the tower is \(60^\circ\).
- Point Q, which is 10 m above point P, from where the angle of depression to the foot of the tower is \(30^\circ\).

### Step 2: Draw the Diagram
1. Draw a vertical line representing the tower.
2. Mark point P at the base of the tower.
3. Mark point Q directly above P, 10 m higher.
4. Draw a line from P to the top of the tower, forming a \(60^\circ\) angle with the horizontal.
5. Draw a line from Q to the foot of the tower, forming a \(30^\circ\) angle with the horizontal.

### Step 3: Calculate the Distance from P to the Tower
Using the angle of elevation from point P:
- Let the height of the tower be \(h\).
- The distance from point P to the foot of the tower is \(d\).

From the right triangle formed:
\[ \tan(60^\circ) = \frac{h}{d} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \sqrt{3} = \frac{h}{d} \implies h = d \cdot \sqrt{3} \quad (1) \]

### Step 4: Calculate the Distance from Q to the Tower
Using the angle of depression from point Q:
- The height from Q to the ground is 10 m, and the angle of depression to the foot of the tower is \(30^\circ\).

From the right triangle formed:
\[ \tan(30^\circ) = \frac{10}{d} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{10}{d} \implies d = 10 \sqrt{3} \quad (2) \]

### Step 5: Substitute to Find the Height of the Tower
Now, substitute equation (2) into equation (1):
\[ h = (10 \sqrt{3}) \cdot \sqrt{3} = 10 \cdot 3 = 30 \text{ m} \]

### Conclusion
The height of the tower is \(30\) meters.