The angle of elevation of the top of a building from the foot of the tower is `60^(@)`. After a flight of 15 seconds, the angle of elevation changes to `30^(@)`. If the aeroplane is flying at a constant height of `1500sqrt3 `m find the speed of the plane in km/hr.

Answer

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

### Step 1: Understand the Problem
We are given:
- The height of the airplane (h) = \(1500\sqrt{3}\) m
- The angle of elevation from the foot of the tower to the airplane at the first position = \(60^\circ\)
- The angle of elevation after 15 seconds = \(30^\circ\)

### Step 2: Set Up the Diagram
Let's denote:
- Point A: Foot of the tower
- Point B: Top of the tower (which is at height \(h\))
- Point C: Position of the airplane after 15 seconds
- Point D: Position of the airplane at the first observation

### Step 3: Calculate the Distance AD
Using triangle ADB where:
- \( \tan(60^\circ) = \frac{h}{AD} \)

We know that:
- \( \tan(60^\circ) = \sqrt{3} \)
- \( h = 1500\sqrt{3} \)

So, we can write:
\[ \sqrt{3} = \frac{1500\sqrt{3}}{AD} \]

Cross-multiplying gives:
\[ AD \cdot \sqrt{3} = 1500\sqrt{3} \]
Dividing both sides by \(\sqrt{3}\):
\[ AD = 1500 \text{ m} \]

### Step 4: Calculate the Distance AE
Now, using triangle CAE where:
- \( \tan(30^\circ) = \frac{h}{AE} \)

We know that:
- \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)

So, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{1500\sqrt{3}}{AE} \]

Cross-multiplying gives:
\[ AE = 1500\sqrt{3} \cdot \sqrt{3} = 1500 \cdot 3 = 4500 \text{ m} \]

### Step 5: Calculate the Distance DE
The distance DE (the distance traveled by the airplane in 15 seconds) is:
\[ DE = AE - AD = 4500 \text{ m} - 1500 \text{ m} = 3000 \text{ m} \]

### Step 6: Calculate the Speed of the Airplane
Speed is defined as distance traveled over time taken. Therefore:
\[ \text{Speed} = \frac{DE}{\text{Time}} = \frac{3000 \text{ m}}{15 \text{ s}} = 200 \text{ m/s} \]

### Step 7: Convert Speed to km/hr
To convert from m/s to km/hr, we use the conversion factor:
\[ \text{Speed in km/hr} = \text{Speed in m/s} \times \frac{18}{5} \]
So:
\[ \text{Speed} = 200 \times \frac{18}{5} = 720 \text{ km/hr} \]

### Final Answer
The speed of the airplane is \(720 \text{ km/hr}\).

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