An aeroplane is flying above a horizontal road between the two consecutive kilometer's stones. These stones are on the opposite sides of the aeroplane. The angles of depressionof these stones from the aeroplane are `30^(@)" and "60^(@)`. Find the height of the aeroplane from the road.

Answer

To solve the problem, we need to find the height of the aeroplane from the road based on the angles of depression to two kilometer stones.

### Step-by-Step Solution:

1. **Understand the Problem Setup**:
- Let the height of the aeroplane from the road be \( h \).
- Let the distance from the point directly below the aeroplane to the first stone (let's call it Stone B) be \( x \) kilometers.
- The distance from the point directly below the aeroplane to the second stone (Stone D) will then be \( 1 - x \) kilometers, since the stones are 1 kilometer apart.

2. **Use the Angle of Depression**:
- From the aeroplane to Stone B, the angle of depression is \( 30^\circ \).
- From the aeroplane to Stone D, the angle of depression is \( 60^\circ \).

3. **Set Up the Trigonometric Relationships**:
- For Stone B (angle of depression \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{h}{x} \]
We know \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies h = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

- For Stone D (angle of depression \( 60^\circ \)):
\[ \tan(60^\circ) = \frac{h}{1 - x} \]
We know \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h}{1 - x} \implies h = \sqrt{3}(1 - x) \quad \text{(Equation 2)} \]

4. **Equate the Two Expressions for Height**:
- From Equation 1 and Equation 2, we have:
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(1 - x) \]
- Cross-multiplying gives:
\[ x = 3(1 - x) \]
\[ x = 3 - 3x \]
\[ 4x = 3 \implies x = \frac{3}{4} \]

5. **Substitute Back to Find Height**:
- Now substitute \( x = \frac{3}{4} \) back into either Equation 1 or Equation 2 to find \( h \).
- Using Equation 1:
\[ h = \frac{\frac{3}{4}}{\sqrt{3}} = \frac{3}{4\sqrt{3}} = \frac{3\sqrt{3}}{12} = \frac{\sqrt{3}}{4} \text{ kilometers} \]

### Final Answer:
The height of the aeroplane from the road is \( \frac{\sqrt{3}}{4} \) kilometers.

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