The angle of elevation of the top of a vertical tower from a point A due east of it is `45^@`. The angle of elevation of the top of the same tower from a point B due south of A is `30^@`. If the distance between A and B is `54sqrt2` m then the height of the tower (in metres), is

Options

A. 108
B. `36sqrt(3)`
C. `54sqrt(3)`
D. 54

Answer

To solve the problem step by step, we will use trigonometric principles and the Pythagorean theorem.

### Step 1: Understand the Geometry
Let's denote:
- O: the top of the tower
- P: the base of the tower
- A: the point due east of the tower
- B: the point due south of A

From the problem, we know:
- The angle of elevation from A to O is \(45^\circ\).
- The angle of elevation from B to O is \(30^\circ\).
- The distance between A and B is \(54\sqrt{2}\) m.

### Step 2: Set Up the Triangles
1. In triangle OAP (where A is due east of P):
- Let the height of the tower (OP) be \(h\).
- Let the distance AP (horizontal distance from A to P) be \(x\).
- From the angle of elevation \(45^\circ\):
\[ \tan(45^\circ) = \frac{h}{x} \implies 1 = \frac{h}{x} \implies h = x \]

2. In triangle OBP (where B is due south of A):
- Let the distance BP (horizontal distance from B to P) be \(y\).
- From the angle of elevation \(30^\circ\):
\[ \tan(30^\circ) = \frac{h}{y} \implies \frac{1}{\sqrt{3}} = \frac{h}{y} \implies y = h\sqrt{3} \]

### Step 3: Relate the Distances
From the geometry, the distance AB can be expressed as:
\[ AB = AP + BP = x + y \]
Given that \(AB = 54\sqrt{2}\), we can substitute \(y\):
\[ x + h\sqrt{3} = 54\sqrt{2} \]

### Step 4: Substitute \(h\) with \(x\)
Since we found that \(h = x\), we can substitute \(h\) in the equation:
\[ x + x\sqrt{3} = 54\sqrt{2} \]
Factoring out \(x\):
\[ x(1 + \sqrt{3}) = 54\sqrt{2} \]
Now, solve for \(x\):
\[ x = \frac{54\sqrt{2}}{1 + \sqrt{3}} \]

### Step 5: Rationalize the Denominator
To simplify \(x\):
\[ x = \frac{54\sqrt{2}(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} = \frac{54\sqrt{2}(1 - \sqrt{3})}{1 - 3} = \frac{54\sqrt{2}(1 - \sqrt{3})}{-2} \]
Thus,
\[ x = -27\sqrt{2}(1 - \sqrt{3}) \]

### Step 6: Calculate \(h\)
Since \(h = x\), we have:
\[ h = -27\sqrt{2}(1 - \sqrt{3}) \]
Calculating this gives us the height of the tower.

### Step 7: Final Calculation
We can also find \(h\) directly using the relation:
\[ h^2 + (54\sqrt{2})^2 = (h\sqrt{3})^2 \]
This leads us to:
\[ h^2 + 2916 = 3h^2 \]
Thus,
\[ 2h^2 = 2916 \implies h^2 = 1458 \implies h = \sqrt{1458} = 54 \text{ m} \]

### Final Answer
The height of the tower is **54 meters**.