An aeroplane flying horizontally 900 m above the ground is observed at an elevation of 60°. After 10 seconds, the elevation changes to 30°. The uniform speed of the aeroplane (in km/hr) is

Options

A. 120
B. `180sqrt(3)`
C. `216sqrt(3)`
D. `50sqrt(3)`

Answer

To solve the problem step by step, we will use the information given about the aeroplane's height and the angles of elevation observed.

### Step 1: Understand the Problem
The aeroplane is flying at a height of 900 m above the ground. The angles of elevation from the observer to the aeroplane are 60° and 30° at two different times.

### Step 2: Set Up the Diagram
Let:
- Point A be the position of the observer on the ground.
- Point P be the position of the aeroplane when the angle of elevation is 60°.
- Point P' be the position of the aeroplane after 10 seconds when the angle of elevation is 30°.

### Step 3: Calculate the Horizontal Distances
Using trigonometry, we can find the horizontal distances from the observer to the points P and P'.

1. **For angle of elevation 60°**:
\[ \tan(60°) = \frac{\text{Height}}{\text{Distance from A to P}} \implies \sqrt{3} = \frac{900}{d_1} \]
Rearranging gives:
\[ d_1 = \frac{900}{\sqrt{3}} = 300\sqrt{3} \text{ m} \]

2. **For angle of elevation 30°**:
\[ \tan(30°) = \frac{\text{Height}}{\text{Distance from A to P'}} \implies \frac{1}{\sqrt{3}} = \frac{900}{d_2} \]
Rearranging gives:
\[ d_2 = 900\sqrt{3} \text{ m} \]

### Step 4: Calculate the Distance Travelled
The distance travelled by the aeroplane in 10 seconds is the difference between the two horizontal distances:
\[ \text{Distance travelled} = d_2 - d_1 = 900\sqrt{3} - 300\sqrt{3} = 600\sqrt{3} \text{ m} \]

### Step 5: Calculate the Speed
Speed is defined as distance travelled over time. The time taken is 10 seconds.
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{600\sqrt{3}}{10} = 60\sqrt{3} \text{ m/s} \]

### Step 6: Convert Speed to km/hr
To convert from m/s to km/hr, we multiply by \( \frac{18}{5} \):
\[ \text{Speed in km/hr} = 60\sqrt{3} \times \frac{18}{5} = \frac{1080\sqrt{3}}{5} \text{ km/hr} \]

### Step 7: Final Calculation
Calculating \( \frac{1080\sqrt{3}}{5} \):
\[ \text{Speed} = 216\sqrt{3} \text{ km/hr} \]

### Conclusion
The uniform speed of the aeroplane is \( 216\sqrt{3} \) km/hr.

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