A flagstaff of length l is fixed on the top of a tower of height h. The angles of elevation of the top and bottom of the flagstaff at a point on the ground are `60^(@)` and `30^(@)` respectively. Then

Options

A. I = 2h
B. 2I=h
C. I=3h
D. 3I=h

Answer

To solve the problem step by step, we will use trigonometric relationships involving the angles of elevation and the heights of the tower and flagstaff.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the height of the tower be \( h \).
- Let the length of the flagstaff be \( l \).
- The total height from the ground to the top of the flagstaff is \( h + l \).
- The angle of elevation to the top of the flagstaff is \( 60^\circ \) and to the bottom of the flagstaff (top of the tower) is \( 30^\circ \).

2. **Using the Angle of Elevation to the Bottom of the Flagstaff**:
- From the point on the ground, let the distance from the base of the tower to the point be \( x \).
- Using the tangent of the angle of elevation to the bottom of the flagstaff:
\[ \tan(30^\circ) = \frac{h}{x} \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \]
- Rearranging gives:
\[ x = h \sqrt{3} \quad \text{(Equation 1)} \]

3. **Using the Angle of Elevation to the Top of the Flagstaff**:
- Now, using the tangent of the angle of elevation to the top of the flagstaff:
\[ \tan(60^\circ) = \frac{h + l}{x} \]
- We know that \( \tan(60^\circ) = \sqrt{3} \), so we can write:
\[ \sqrt{3} = \frac{h + l}{x} \]
- Substituting \( x \) from Equation 1:
\[ \sqrt{3} = \frac{h + l}{h \sqrt{3}} \]
- Cross-multiplying gives:
\[ \sqrt{3} \cdot h \sqrt{3} = h + l \]
- Simplifying this:
\[ 3h = h + l \]
- Rearranging gives:
\[ l = 3h - h = 2h \quad \text{(Equation 2)} \]

4. **Conclusion**:
- From Equation 2, we find that:
\[ l = 2h \]

### Final Answer:
The relationship between the height of the tower and the length of the flagstaff is:
\[ l = 2h \]