The angle of elevation of the loop of a vertical tower standing on a horizontal plane is observed to be `45^(@)` from a point `A` on the plane. Let `B` be the point `30m` vertically above the point `A`. If the angle of elevation of the top of the tower from `B` be `30^(@)`, then the distance (in m) of the foot of the lower from the point `A` is:

Options

A. `15 (3 +sqrt(3))`
B. `15 (5-sqrt(3))`
C. `15 (3-sqrt(3))`
D. `15(1+sqrt(3))`

Answer

To solve the problem step by step, we will analyze the situation using trigonometric principles and geometry.

### Step 1: Understand the Geometry
We have a vertical tower, and we denote the following points:
- Let \( O \) be the foot of the tower on the ground.
- Let \( A \) be the point on the ground where the angle of elevation to the top of the tower is \( 45^\circ \).
- Let \( B \) be the point \( 30 \, m \) vertically above point \( A \).
- Let \( C \) be the top of the tower.

### Step 2: Set Up the Variables
Let:
- \( h \) be the height of the tower.
- \( x \) be the horizontal distance from point \( A \) to point \( O \).

### Step 3: Use the Angle of Elevation from Point A
From point \( A \), the angle of elevation to the top of the tower \( C \) is \( 45^\circ \). Therefore, we can use the tangent function:
\[ \tan(45^\circ) = \frac{h}{x} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{h}{x} \implies h = x \]

### Step 4: Use the Angle of Elevation from Point B
From point \( B \), the angle of elevation to the top of the tower \( C \) is \( 30^\circ \). The height from point \( B \) to the top of the tower is \( h - 30 \) (since \( B \) is \( 30 \, m \) above point \( A \)). Therefore, we can write:
\[ \tan(30^\circ) = \frac{h - 30}{x} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h - 30}{x} \]
Cross-multiplying gives:
\[ x = \sqrt{3}(h - 30) \]

### Step 5: Substitute \( h \) from Step 3 into Step 4
We already found that \( h = x \). Substitute this into the equation:
\[ x = \sqrt{3}(x - 30) \]

### Step 6: Solve for \( x \)
Expanding the equation:
\[ x = \sqrt{3}x - 30\sqrt{3} \]
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 7: Rationalize the Denominator
To rationalize the denominator:
\[ x = \frac{-30\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{-30\sqrt{3}(1 + \sqrt{3})}{1 - 3} = \frac{-30\sqrt{3}(1 + \sqrt{3})}{-2} \]
This simplifies to:
\[ x = 15\sqrt{3}(1 + \sqrt{3}) = 15(3 + \sqrt{3}) \]

### Final Answer
Thus, the distance of the foot of the tower from point \( A \) is:
\[ x = 15(3 + \sqrt{3}) \, m \]