The angle of elevation of an aeroplane from a point on the ground is `60^(@)` . After flying for 30 seconds, the angle of elevation charges to `30^(@)` . If the aeroplane is flying at a height of 4500 m, then what is the speed (in m/s) of aeroplane

Options

A. `50 sqrt""3`
B. `100 sqrt""3`
C. `200 sqrt""3`
D. `300 sqrt""3`

Answer

To solve the problem step by step, we will use trigonometric principles to find the speed of the airplane.

### Step 1: Understand the problem
We have an airplane flying at a height of 4500 m. The angle of elevation from a point on the ground to the airplane is initially 60 degrees and changes to 30 degrees after the airplane has flown for 30 seconds. We need to find the speed of the airplane.

### Step 2: Set up the triangles
Let:
- Point A be the point on the ground directly below the airplane when the angle of elevation is 60 degrees.
- Point B be the position of the airplane when the angle of elevation is 60 degrees.
- Point C be the position of the airplane when the angle of elevation is 30 degrees.
- Point D be the point on the ground directly below the airplane when the angle of elevation is 30 degrees.

### Step 3: Calculate distance AD using triangle ADB
In triangle ADB, we can use the tangent function:
\[ \tan(60^\circ) = \frac{\text{Height}}{\text{Base}} = \frac{4500}{AD} \]
Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{4500}{AD} \]
Rearranging gives:
\[ AD = \frac{4500}{\sqrt{3}} = 1500\sqrt{3} \text{ m} \]

### Step 4: Calculate distance CE using triangle CAE
In triangle CAE, we can use the tangent function again:
\[ \tan(30^\circ) = \frac{\text{Height}}{\text{Base}} = \frac{4500}{CE} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{1}{\sqrt{3}} = \frac{4500}{CE} \]
Rearranging gives:
\[ CE = 4500\sqrt{3} \text{ m} \]

### Step 5: Calculate the distance covered by the airplane
The distance covered by the airplane (DE) is the difference between CE and AD:
\[ DE = CE - AD = 4500\sqrt{3} - 1500\sqrt{3} = (4500 - 1500)\sqrt{3} = 3000\sqrt{3} \text{ m} \]

### Step 6: Calculate the speed of the airplane
Speed is defined as distance divided by time. The time given is 30 seconds:
\[ \text{Speed} = \frac{DE}{\text{Time}} = \frac{3000\sqrt{3}}{30} = 100\sqrt{3} \text{ m/s} \]

### Final Answer
The speed of the airplane is \(100\sqrt{3}\) m/s.

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