A vertical pole and a vertical tower are standing on the same level ground .Height of the pole is 10 metres .From the top of the pole the angle of elevation of the top of the tower and angle of depresion of the foot of the tower are `60^(@)and30^(@)` respectively . The height of the tower is

Options

A. a) 20 m
B. b) 30 m
C. c) 40 m
D. d) 50 m

Answer

To solve the problem, we will use trigonometric ratios. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a vertical pole of height 10 meters and a vertical tower. From the top of the pole, the angle of elevation to the top of the tower is 60 degrees, and the angle of depression to the foot of the tower is 30 degrees. We need to find the height of the tower.

### Step 2: Draw the Diagram
Let's label the points:
- Let A be the top of the pole.
- Let B be the foot of the pole.
- Let C be the top of the tower.
- Let D be the foot of the tower.

The height of the pole (AB) is 10 meters. We need to find the height of the tower (AC).

### Step 3: Set Up the Triangles
1. **Triangle ABE** (where E is the foot of the tower):
- Angle ABE = 60 degrees (angle of elevation)
- AB = 10 meters (height of the pole)
- Let BE = x (the distance from the foot of the pole to the foot of the tower).

Using the tangent function:
\[ \tan(60^\circ) = \frac{AC - AB}{BE} \]
This gives us:
\[ \sqrt{3} = \frac{AC - 10}{x} \quad \text{(1)} \]

2. **Triangle BEC**:
- Angle CBE = 30 degrees (angle of depression)
- BC = 10 meters (height of the pole)
- BE = x (the distance from the foot of the pole to the foot of the tower).

Using the tangent function:
\[ \tan(30^\circ) = \frac{BC}{BE} \]
This gives us:
\[ \frac{1}{\sqrt{3}} = \frac{10}{x} \quad \text{(2)} \]

### Step 4: Solve for x
From equation (2):
\[ x = 10\sqrt{3} \]

### Step 5: Substitute x in Equation (1)
Now substitute the value of x into equation (1):
\[ \sqrt{3} = \frac{AC - 10}{10\sqrt{3}} \]
Cross-multiplying gives:
\[ \sqrt{3} \cdot 10\sqrt{3} = AC - 10 \]
\[ 30 = AC - 10 \]
\[ AC = 40 \text{ meters} \]

### Conclusion
The height of the tower (AC) is **40 meters**.

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