The angle of elevation of the top of a pillar from the foot and the top of a building 20 m high , are `60^(@)and30^(@)` respectively . The height of the pillar is

Options

A. 10 m
B. `10sqrt(3)`
C. 60 m
D. 30 m

Answer

To solve the problem, we need to determine the height of the pillar based on the angles of elevation from two different points: the foot of the pillar and the top of a 20-meter high building.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the height of the pillar be \( h \).
- The height of the building is given as \( 20 \) m.
- The angle of elevation from the foot of the pillar to the top of the pillar is \( 60^\circ \).
- The angle of elevation from the top of the building to the top of the pillar is \( 30^\circ \).

2. **Setting Up the Diagram**:
- Let point \( A \) be the top of the pillar, point \( B \) be the foot of the pillar, and point \( C \) be the top of the building.
- The height \( AB = h \) (height of the pillar).
- The height \( BC = 20 \) m (height of the building).
- The distance from the foot of the building to the foot of the pillar is \( x \).

3. **Using the First Triangle (from the foot of the pillar)**:
- In triangle \( ABC \), using the angle of elevation \( 60^\circ \):
\[ \tan(60^\circ) = \frac{h}{x} \]
- Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{h}{x} \implies h = x\sqrt{3} \quad \text{(Equation 1)} \]

4. **Using the Second Triangle (from the top of the building)**:
- In triangle \( ACD \), using the angle of elevation \( 30^\circ \):
\[ \tan(30^\circ) = \frac{h - 20}{x} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h - 20}{x} \implies h - 20 = \frac{x}{\sqrt{3}} \quad \text{(Equation 2)} \]

5. **Substituting Equation 1 into Equation 2**:
- Substitute \( h = x\sqrt{3} \) into Equation 2:
\[ x\sqrt{3} - 20 = \frac{x}{\sqrt{3}} \]
- Multiply through by \( \sqrt{3} \) to eliminate the fraction:
\[ 3x - 20\sqrt{3} = x \]
- Rearranging gives:
\[ 3x - x = 20\sqrt{3} \implies 2x = 20\sqrt{3} \implies x = 10\sqrt{3} \]

6. **Finding the Height of the Pillar**:
- Now substitute \( x \) back into Equation 1 to find \( h \):
\[ h = x\sqrt{3} = 10\sqrt{3} \cdot \sqrt{3} = 10 \cdot 3 = 30 \text{ m} \]

### Final Answer:
The height of the pillar is \( 30 \) meters.