The angle of elevation of the top of a tower from two points A and B lying on the horizontal through the foot of the tower are respectively `15^(@)and30^(@)` .If A and B are on the same side of the tower and AB = 48 metre , then the height of the tower is :

Options

A. `24sqrt(3)` metre
B. 24 metre
C. `24sqrt(2)` metre
D. 96 metre

Answer

To find the height of the tower based on the given angles of elevation from points A and B, we can follow these steps:

### Step 1: Understand the Problem
We have two points A and B from which the angles of elevation to the top of the tower are given as \(15^\circ\) and \(30^\circ\) respectively. The distance between points A and B is 48 meters.

### Step 2: Set Up the Diagram
Let:
- \(H\) be the height of the tower.
- \(x\) be the horizontal distance from point B to the foot of the tower.
- Therefore, the distance from point A to the foot of the tower will be \(x + 48\).

### Step 3: Use Trigonometric Ratios
From point B (angle \(30^\circ\)):
\[ \tan(30^\circ) = \frac{H}{x} \]
Using the value of \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{H}{x} \implies H = \frac{x}{\sqrt{3}} \tag{1} \]

From point A (angle \(15^\circ\)):
\[ \tan(15^\circ) = \frac{H}{x + 48} \]
Using the value of \(\tan(15^\circ) = 2 - \sqrt{3}\):
\[ 2 - \sqrt{3} = \frac{H}{x + 48} \implies H = (2 - \sqrt{3})(x + 48) \tag{2} \]

### Step 4: Equate the Two Expressions for Height
From equations (1) and (2):
\[ \frac{x}{\sqrt{3}} = (2 - \sqrt{3})(x + 48) \]

### Step 5: Solve for \(x\)
Multiply both sides by \(\sqrt{3}\):
\[ x = \sqrt{3}(2 - \sqrt{3})(x + 48) \]
Expanding the right side:
\[ x = (2\sqrt{3} - 3)(x + 48) \]
Distributing:
\[ x = (2\sqrt{3} - 3)x + 48(2\sqrt{3} - 3) \]
Rearranging:
\[ x - (2\sqrt{3} - 3)x = 48(2\sqrt{3} - 3) \]
Factoring out \(x\):
\[ x(1 - (2\sqrt{3} - 3)) = 48(2\sqrt{3} - 3) \]
This simplifies to:
\[ x(4 - 2\sqrt{3}) = 48(2\sqrt{3} - 3) \]
Thus:
\[ x = \frac{48(2\sqrt{3} - 3)}{4 - 2\sqrt{3}} \tag{3} \]

### Step 6: Substitute \(x\) Back to Find \(H\)
Now substitute \(x\) from equation (3) back into equation (1) to find \(H\):
\[ H = \frac{x}{\sqrt{3}} = \frac{48(2\sqrt{3} - 3)}{(4 - 2\sqrt{3})\sqrt{3}} \]

### Step 7: Calculate the Height
Calculating \(H\):
\[ H = \frac{48(2\sqrt{3} - 3)}{(4 - 2\sqrt{3})\sqrt{3}} = 24 \text{ meters} \]

### Final Answer
The height of the tower is **24 meters**.

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