A man observes the elevation of a balloon to be `30^(@)` . He, then walks 1 km towards the balloon and finds that the elevation is `60^(@)`. What is the height of the balloon?

Options

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

Explanation

To solve the problem step by step, we will use trigonometric ratios and the information provided in the question.

### Step 1: Understand the Problem
We have a man observing a balloon at two different points. The first observation is at an angle of elevation of \(30^\circ\) and the second observation, after walking 1 km towards the balloon, is at an angle of elevation of \(60^\circ\). We need to find the height of the balloon.

### Step 2: Set Up the Diagram
Let:
- \(AB\) be the height of the balloon.
- \(D\) be the initial position of the man.
- \(C\) be the position of the man after walking 1 km towards the balloon.
- \(BD\) be the height of the balloon, which we denote as \(h\).
- \(AD\) be the horizontal distance from the man to the base of the balloon at point \(B\) before he walks.
- \(CD\) is the distance the man walks towards the balloon, which is 1 km.

### Step 3: Use the First Angle of Elevation
From point \(D\) (initial position), the angle of elevation is \(30^\circ\):
\[ \tan(30^\circ) = \frac{h}{AD} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{AD} \]
This gives us:
\[ AD = h \cdot \sqrt{3} \]

### Step 4: Use the Second Angle of Elevation
After walking 1 km towards the balloon, from point \(C\), the angle of elevation is \(60^\circ\):
\[ \tan(60^\circ) = \frac{h}{AD - 1} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \sqrt{3} = \frac{h}{AD - 1} \]
This gives us:
\[ AD - 1 = \frac{h}{\sqrt{3}} \]
Thus:
\[ AD = \frac{h}{\sqrt{3}} + 1 \]

### Step 5: Set the Two Expressions for AD Equal
Now we have two expressions for \(AD\):
1. \(AD = h \cdot \sqrt{3}\)
2. \(AD = \frac{h}{\sqrt{3}} + 1\)

Setting them equal gives:
\[ h \cdot \sqrt{3} = \frac{h}{\sqrt{3}} + 1 \]

### Step 6: Solve for h
To eliminate the fractions, multiply the entire equation by \(\sqrt{3}\):
\[ 3h = h + \sqrt{3} \]
Rearranging gives:
\[ 3h - h = \sqrt{3} \]
\[ 2h = \sqrt{3} \]
Thus:
\[ h = \frac{\sqrt{3}}{2} \text{ km} \]

### Step 7: Final Answer
The height of the balloon is:
\[ h = \frac{\sqrt{3}}{2} \text{ km} \approx 0.866 \text{ km} \]