The angle of elevation of an aeroplane from a point on the ground is `45^@`. After a flight of 10 sec, the elevation changes to `30^@`. If the aeroplane is flying at a height of 3 km, then find the speed of the aeroplane. (Use `sqrt3 = 1.732`)

Options

A. 785.46 km/hr
B. 790.56 km/hr
C. 780.56 km/hr
D. 782.65 km/hr

Answer

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

### Step 1: Understand the situation
We have an aeroplane flying at a height of 3 km. The angle of elevation from a point on the ground changes from 45° to 30° after 10 seconds.

### Step 2: Draw the diagram
- Let point A be the position of the aeroplane when the angle of elevation is 45°.
- Let point B be the point on the ground directly below the aeroplane when the angle of elevation is 45°.
- Let point C be the position of the aeroplane after 10 seconds when the angle of elevation is 30°.
- Let point D be the point on the ground directly below the aeroplane when the angle of elevation is 30°.

### Step 3: Use triangle APB (when angle is 45°)
In triangle APB:
- The height (opposite side) = 3 km (height of the aeroplane)
- The angle of elevation = 45°

Using the tangent function:
\[ \tan(45°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{AP} \]
Since \(\tan(45°) = 1\):
\[ 1 = \frac{3}{AP} \implies AP = 3 \text{ km} \]

### Step 4: Use triangle PCD (when angle is 30°)
In triangle PCD:
- The height (opposite side) = 3 km
- The angle of elevation = 30°

Using the tangent function:
\[ \tan(30°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{PC} \]
Since \(\tan(30°) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{3}{PC} \implies PC = 3\sqrt{3} \text{ km} \]

### Step 5: Calculate the distance PC
Using \(\sqrt{3} \approx 1.732\):
\[ PC = 3 \times 1.732 = 5.196 \text{ km} \]

### Step 6: Find the distance AC
Now, we can find the distance AC (the distance travelled by the aeroplane in 10 seconds):
\[ AC = PC - AP = 5.196 \text{ km} - 3 \text{ km} = 2.196 \text{ km} \]

### Step 7: Calculate the speed of the aeroplane
The distance AC is covered in 10 seconds. To find the speed in km/h:
1. First, find the speed in km/s:
\[ \text{Speed} = \frac{2.196 \text{ km}}{10 \text{ s}} = 0.2196 \text{ km/s} \]
2. Convert km/s to km/h (1 hour = 3600 seconds):
\[ \text{Speed} = 0.2196 \text{ km/s} \times 3600 \text{ s/h} = 790.56 \text{ km/h} \]

### Final Answer
The speed of the aeroplane is **790.56 km/h**.

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