The angular elevation of the top of a tower from a distant point on the horizontal ground is observed to be `30^(@)and` proceeding 30 metre from the point towards the foot of the tower it is observed to be `45^(@)` .Find the height of the tower .

Options

A. 15 metre
B. `15sqrt(3)` metre
C. `15(sqrt(3)+1)` metre
D. None of these

Answer

To find the height of the tower based on the given angles of elevation, we can follow these steps:

### Step 1: Define the problem
Let:
- \( H \) = height of the tower
- \( X \) = distance from the first point of observation to the base of the tower
- The first observation point is at a distance \( X \) from the base of the tower where the angle of elevation is \( 30^\circ \).
- After moving 30 meters closer to the tower, the new distance to the base becomes \( X - 30 \) where the angle of elevation is \( 45^\circ \).

### Step 2: Set up the first triangle (30-degree angle)
Using the tangent function for the first observation:
\[ \tan(30^\circ) = \frac{H}{X} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{H}{X} \]
This implies:
\[ H = \frac{X}{\sqrt{3}} \quad \text{(1)} \]

### Step 3: Set up the second triangle (45-degree angle)
Using the tangent function for the second observation:
\[ \tan(45^\circ) = \frac{H}{X - 30} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{H}{X - 30} \]
This implies:
\[ H = X - 30 \quad \text{(2)} \]

### Step 4: Equate the two expressions for \( H \)
From equations (1) and (2):
\[ \frac{X}{\sqrt{3}} = X - 30 \]

### Step 5: Solve for \( X \)
To eliminate \( X \), multiply both sides by \( \sqrt{3} \):
\[ X = \sqrt{3}(X - 30) \]
Expanding the right side:
\[ X = \sqrt{3}X - 30\sqrt{3} \]
Rearranging gives:
\[ X - \sqrt{3}X = -30\sqrt{3} \]
Factoring out \( X \):
\[ X(1 - \sqrt{3}) = -30\sqrt{3} \]
Thus:
\[ X = \frac{-30\sqrt{3}}{1 - \sqrt{3}} \]

### Step 6: Rationalize the denominator
Multiply numerator and denominator by \( 1 + \sqrt{3} \):
\[ X = \frac{-30\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \]
The denominator simplifies to:
\[ 1 - 3 = -2 \]
So:
\[ X = \frac{30\sqrt{3}(1 + \sqrt{3})}{2} \]
This simplifies to:
\[ X = 15\sqrt{3}(1 + \sqrt{3}) = 15\sqrt{3} + 45 \]

### Step 7: Substitute back to find \( H \)
Using \( H = X - 30 \):
\[ H = (15\sqrt{3} + 45) - 30 \]
Thus:
\[ H = 15\sqrt{3} + 15 \]

### Final Answer
The height of the tower is:
\[ H = 15(\sqrt{3} + 1) \text{ meters} \]