If the angles of elevation of a ballon from the two consecultive kilometre stones along a road are `30^@` and `60^@` respectively, then the height of the balloon above the ground will be

Options

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

Answer

To find the height of the balloon above the ground based on the angles of elevation from two consecutive kilometer stones, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Problem Setup**:
- Let the height of the balloon be \( H \).
- Let the distance from the first kilometer stone (point A) to the point directly below the balloon (point B) be \( Y \).
- The distance between the two kilometer stones is 1 km.

2. **Draw the Right Triangles**:
- From point A (first kilometer stone), the angle of elevation to the balloon is \( 30^\circ \).
- From point B (second kilometer stone, which is 1 km away from A), the angle of elevation to the balloon is \( 60^\circ \).

3. **Set Up the First Triangle (Triangle ABD)**:
- In triangle ABD, where:
- \( AB = H \) (height of the balloon),
- \( BD = Y + 1 \) (distance from point B to the point directly below the balloon).
- Using the tangent of the angle:
\[ \tan(30^\circ) = \frac{AB}{BD} = \frac{H}{Y + 1} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{H}{Y + 1} = \frac{1}{\sqrt{3}} \implies H = \frac{Y + 1}{\sqrt{3}} \quad \text{(Equation 1)} \]

4. **Set Up the Second Triangle (Triangle ABC)**:
- In triangle ABC, where:
- \( AB = H \),
- \( BC = Y \).
- Using the tangent of the angle:
\[ \tan(60^\circ) = \frac{AB}{BC} = \frac{H}{Y} \]
- Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \frac{H}{Y} = \sqrt{3} \implies H = \sqrt{3}Y \quad \text{(Equation 2)} \]

5. **Equate the Two Equations**:
- From Equation 1: \( H = \frac{Y + 1}{\sqrt{3}} \)
- From Equation 2: \( H = \sqrt{3}Y \)
- Setting them equal:
\[ \frac{Y + 1}{\sqrt{3}} = \sqrt{3}Y \]

6. **Solve for Y**:
- Cross-multiplying gives:
\[ Y + 1 = 3Y \]
- Rearranging:
\[ 1 = 3Y - Y \implies 1 = 2Y \implies Y = \frac{1}{2} \]

7. **Substitute Y Back to Find H**:
- Substitute \( Y = \frac{1}{2} \) into Equation 2:
\[ H = \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \]

8. **Final Answer**:
- The height of the balloon above the ground is \( \frac{\sqrt{3}}{2} \) km.