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

Options

A. `1//2km `
B. `sqrt3 //2 km`
C. `1//3 km`
D. `1 km`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
A man observes a balloon at an angle of elevation of 30 degrees. After walking 1 km towards the balloon, the angle of elevation changes to 60 degrees. We need to find the height of the balloon.

### Step 2: Draw a Diagram
1. Let point A be the position of the balloon.
2. Let point B be the position of the man when he first observes the balloon.
3. Let point C be the position of the man after he walks 1 km towards the balloon.
4. The height of the balloon is represented as AB (h), and the horizontal distance from the man to the base of the balloon is represented as BC (a).

### Step 3: Set Up the First Triangle (Triangle ADB)
In triangle ADB, we have:
- Angle DAB = 30 degrees
- Using the tangent function:
\[ \tan(30^\circ) = \frac{h}{a + 1} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h}{a + 1} \]
Rearranging gives:
\[ a + 1 = \sqrt{3}h \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Triangle (Triangle ABC)
In triangle ABC, we have:
- Angle CAB = 60 degrees
- Using the tangent function:
\[ \tan(60^\circ) = \frac{h}{a} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we can write:
\[ \sqrt{3} = \frac{h}{a} \]
Rearranging gives:
\[ h = \sqrt{3}a \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 2 into Equation 1
Now, substitute \(h\) from Equation 2 into Equation 1:
\[ a + 1 = \sqrt{3}(\sqrt{3}a) \]
This simplifies to:
\[ a + 1 = 3a \]
Rearranging gives:
\[ 1 = 3a - a \]
\[ 1 = 2a \]
\[ a = \frac{1}{2} \text{ km} \]

### Step 6: Find the Height of the Balloon
Now substitute \(a\) back into Equation 2 to find \(h\):
\[ h = \sqrt{3} \left(\frac{1}{2}\right) \]
\[ h = \frac{\sqrt{3}}{2} \text{ km} \]

### Final Answer
The height of the balloon is \(\frac{\sqrt{3}}{2}\) km.

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