The angles of elevation of top and bottom of a flag kept on a flagpost from 30 metres distance are `45^(@)and30^(@)` respectively .Height of the flat is (taking `sqrt(3)=1.732`)

Options

A. `12sqrt(3)` metre
B. 15 metre
C. 14 .32 metre
D. 12. 68 metre

Answer

To solve the problem of finding the height of the flag on the flagpost, we can break it down into steps using trigonometric principles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a flagpole with a flag on top. We are given two angles of elevation from a point 30 meters away from the base of the flagpole:
- The angle of elevation to the top of the flag is \(45^\circ\).
- The angle of elevation to the bottom of the flag (the top of the flagpole) is \(30^\circ\).

2. **Setting Up the Triangles**:
Let:
- \(A\) be the point where the observer is standing (30 meters away from the flagpole).
- \(B\) be the top of the flag.
- \(C\) be the bottom of the flag (top of the flagpole).
- \(D\) be the ground level at the base of the flagpole.

We can form two right triangles:
- Triangle \(ABD\) for the angle of elevation to the top of the flag.
- Triangle \(ACD\) for the angle of elevation to the bottom of the flag.

3. **Using Trigonometry for Triangle \(ABD\)**:
For triangle \(ABD\) (where \( \angle ADB = 45^\circ \)):
\[ \tan(45^\circ) = \frac{AB}{AD} \]
Since \(AD = 30\) meters and \(\tan(45^\circ) = 1\):
\[ 1 = \frac{AB}{30} \implies AB = 30 \text{ meters} \]

4. **Using Trigonometry for Triangle \(ACD\)**:
For triangle \(ACD\) (where \( \angle ACD = 30^\circ \)):
\[ \tan(30^\circ) = \frac{AC}{AD} \]
We know \(AD = 30\) meters and \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{AC}{30} \implies AC = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ meters} \]

5. **Calculating the Height of the Flag**:
The height of the flag \(h\) is the difference between the height of the top of the flag \(AB\) and the height of the bottom of the flag \(AC\):
\[ h = AB - AC = 30 - 10\sqrt{3} \]
Substituting \(\sqrt{3} \approx 1.732\):
\[ h = 30 - 10 \times 1.732 = 30 - 17.32 = 12.68 \text{ meters} \]

### Final Answer:
The height of the flag is **12.68 meters**.