The angle of elevation of an aeroplane from a point on the ground is `60^(@)` .After flying for 30 seconds ,the angle of elevation changes to `30^(@)` .If the aeroplane is flying at a height of `1500sqrt(3)` metre , then what is the speed (in m/s) of aeroplane?

Options

A. 50
B. 100
C. 200
D. 300

Answer

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

### Step 1: Understand the Problem
We have an aeroplane flying at a height of \( 1500\sqrt{3} \) meters. The angle of elevation from a point on the ground to the aeroplane changes from \( 60^\circ \) to \( 30^\circ \) after 30 seconds. We need to find the speed of the aeroplane in meters per second.

### Step 2: Set Up the Diagram
Let's denote:
- Point A: The observer on the ground.
- Point B: The position of the aeroplane when the angle of elevation is \( 60^\circ \).
- Point C: The position of the aeroplane after 30 seconds when the angle of elevation is \( 30^\circ \).
- BD and CE: The height of the aeroplane, which is \( 1500\sqrt{3} \) meters.

### Step 3: Calculate the Distance from Point A to Point B (AD)
In triangle ABD:
- We have \( \tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BD}{AD} \).
- Given \( BD = 1500\sqrt{3} \) meters, we can write:
\[ \tan(60^\circ) = \sqrt{3} \implies \sqrt{3} = \frac{1500\sqrt{3}}{AD} \]
- Rearranging gives:
\[ AD = \frac{1500\sqrt{3}}{\sqrt{3}} = 1500 \text{ meters} \]

### Step 4: Calculate the Distance from Point A to Point C (AE)
In triangle CAE:
- We have \( \tan(30^\circ) = \frac{CE}{AE} \).
- Again, \( CE = 1500\sqrt{3} \) meters, so:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \implies \frac{1}{\sqrt{3}} = \frac{1500\sqrt{3}}{AE} \]
- Rearranging gives:
\[ AE = 1500\sqrt{3} \cdot \sqrt{3} = 1500 \cdot 3 = 4500 \text{ meters} \]

### Step 5: Calculate the Distance Travelled by the Aeroplane
The distance travelled by the aeroplane from point B to point C is:
\[ \text{Distance} = AE - AD = 4500 - 1500 = 3000 \text{ meters} \]

### Step 6: Calculate the Speed of the Aeroplane
The speed of the aeroplane is given by the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{3000 \text{ meters}}{30 \text{ seconds}} = 100 \text{ m/s} \]

### Final Answer
The speed of the aeroplane is \( 100 \text{ m/s} \).

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