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 principles related to right triangles formed by the tower, flagstaff, and the point of observation on the ground.

### Step 1: Define the problem
Let:
- \( h \) = height of the tower
- \( L \) = length of the flagstaff
- \( A \) = point on the ground where the angles are measured
- \( B \) = top of the flagstaff
- \( C \) = bottom of the flagstaff (top of the tower)
- \( D \) = ground level directly below the flagstaff

From the problem, we know:
- The angle of elevation to point \( B \) (top of the flagstaff) is \( 60^\circ \).
- The angle of elevation to point \( C \) (bottom of the flagstaff) is \( 30^\circ \).

### Step 2: Set up the equations using trigonometry
Using the tangent function for the angles of elevation:

1. For triangle \( ABD \) (where \( AB \) is the flagstaff and \( AD \) is the distance from point \( A \) to the base of the tower):
\[ \tan(60^\circ) = \frac{h + L}{AD} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{h + L}{AD} \quad \text{(Equation 1)} \]

2. For triangle \( ACD \) (where \( AC \) is the height of the tower):
\[ \tan(30^\circ) = \frac{h}{AD} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{AD} \quad \text{(Equation 2)} \]

### Step 3: Solve for \( AD \) from Equation 2
From Equation 2, we can express \( AD \) in terms of \( h \):
\[ AD = \sqrt{3}h \quad \text{(Equation 3)} \]

### Step 4: Substitute \( AD \) into Equation 1
Now substitute Equation 3 into Equation 1:
\[ \sqrt{3} = \frac{h + L}{\sqrt{3}h} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ \sqrt{3} \cdot \sqrt{3}h = h + L \]
\[ 3h = h + L \]
Now, isolate \( L \):
\[ L = 3h - h = 2h \]

### Conclusion
Thus, we have derived that:
\[ L = 2h \]

### Final Answer
The correct option is:
**Option 1: \( L = 2h \)**