ABCDEF is a regular polygon .Two poles at C and D are standing vertically and subtend angles of elevation `30^(@)and60^(@)` at A respectively .What is the ratio of the heigth of the pole at C to that of the pole at D ?

Options

A. `1:1`
B. `1:2sqrt(3)`
C. `2sqrt(3):1`
D. `2:sqrt(3)`

Answer

To solve the problem, we need to find the ratio of the height of the pole at point C (h1) to the height of the pole at point D (h2). We will use trigonometric relationships based on the angles of elevation given.

### Step-by-Step Solution:

1. **Identify the Angles of Elevation**:
- The angle of elevation from point A to pole C is 30 degrees.
- The angle of elevation from point A to pole D is 60 degrees.

2. **Set Up the Geometry**:
- Let the distance from point A to point C be \( AC = x \).
- Let the distance from point A to point D be \( AD = y \).
- The height of the pole at C is \( h_1 \) and at D is \( h_2 \).

3. **Use Trigonometric Ratios**:
- For pole C:
\[ \tan(30^\circ) = \frac{h_1}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h_1}{x} \implies h_1 = x \cdot \frac{1}{\sqrt{3}} \implies h_1 = \frac{x}{\sqrt{3}} \]

- For pole D:
\[ \tan(60^\circ) = \frac{h_2}{y} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h_2}{y} \implies h_2 = y \cdot \sqrt{3} \implies h_2 = y \sqrt{3} \]

4. **Determine the Distances**:
- Since ABCDEF is a regular polygon, the distances \( AC \) and \( AD \) can be expressed in terms of the side length of the polygon. Let’s assume the side length is \( a \).
- For a regular hexagon, \( AC = a \) and \( AD = 2a \) (since D is two sides away from A).

5. **Substitute the Distances**:
- Substitute \( x = a \) and \( y = 2a \) into the height equations:
\[ h_1 = \frac{a}{\sqrt{3}} \quad \text{and} \quad h_2 = 2a \sqrt{3} \]

6. **Find the Ratio**:
- Now, we can find the ratio of the heights:
\[ \frac{h_1}{h_2} = \frac{\frac{a}{\sqrt{3}}}{2a \sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{1}{2\sqrt{3}} = \frac{1}{6} \]

### Final Answer:
The ratio of the height of the pole at C to that of the pole at D is \( \frac{1}{6} \).