A balloon is moving with the wind in a horizontal line at a height of `36sqrt3` m. The angle of elevation of the balloon from a point A on the ground is `60^@`. After some time, the angle of elevation changes to `30^@`. Find the distance travelled by the balloon.

Options

A. 72 m
B. 78 m
C. 86 m
D. 82 m

Answer

To solve the problem step by step, we will use trigonometric relationships in right triangles formed by the balloon's height and the angles of elevation.

### Step 1: Understand the Problem
We have a balloon at a height of \(36\sqrt{3}\) meters. The angle of elevation from point A to the balloon is initially \(60^\circ\) and later changes to \(30^\circ\). We need to find the horizontal distance travelled by the balloon.

### Step 2: Draw the Diagram
Draw a horizontal line representing the ground. Mark point A on this line. Draw a vertical line from point A to represent the height of the balloon, which is \(36\sqrt{3}\) m. Mark the position of the balloon at two different points corresponding to the angles of elevation \(60^\circ\) and \(30^\circ\).

### Step 3: Set Up the Right Triangles
From point A, draw lines to the balloon at the two angles of elevation:
- For \(60^\circ\), label the point where the balloon is as point B.
- For \(30^\circ\), label the new position of the balloon as point C.

### Step 4: Calculate the Distances
Using the tangent function, we can find the horizontal distances from point A to points B and C.

1. **For angle \(60^\circ\)**:
\[ \tan(60^\circ) = \frac{\text{Height}}{\text{Distance AB}} \Rightarrow \sqrt{3} = \frac{36\sqrt{3}}{d_1} \]
Solving for \(d_1\):
\[ d_1 = \frac{36\sqrt{3}}{\sqrt{3}} = 36 \text{ meters} \]

2. **For angle \(30^\circ\)**:
\[ \tan(30^\circ) = \frac{\text{Height}}{\text{Distance AC}} \Rightarrow \frac{1}{\sqrt{3}} = \frac{36\sqrt{3}}{d_2} \]
Solving for \(d_2\):
\[ d_2 = 36\sqrt{3} \cdot \sqrt{3} = 36 \cdot 3 = 108 \text{ meters} \]

### Step 5: Calculate the Distance Travelled by the Balloon
The distance travelled by the balloon is the difference between the two distances:
\[ \text{Distance travelled} = d_2 - d_1 = 108 - 36 = 72 \text{ meters} \]

### Final Answer
The distance travelled by the balloon is \(72\) meters.

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