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 use trigonometric principles and the properties of right triangles.

### Step 1: Understand the Setup
We have a vertical tower and two points: A on the ground and B, which is 30 meters above A. The angle of elevation from A to the top of the tower is 45 degrees, and from B to the top of the tower, it is 30 degrees.

### Step 2: Define Variables
Let:
- \( PQ \) be the height of the tower.
- \( AR \) be the horizontal distance from point A to the foot of the tower (let's denote it as \( x \)).
- \( BQ \) be the height from point A to point B, which is 30 m.

### Step 3: Use Triangle from Point B
From point B, we can use the angle of elevation to find the relationship between the height of the tower and the horizontal distance:
\[ \tan(30^\circ) = \frac{PQ - 30}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so we can write:
\[ \frac{1}{\sqrt{3}} = \frac{PQ - 30}{x} \]
From this, we can express \( PQ \):
\[ PQ - 30 = \frac{x}{\sqrt{3}} \implies PQ = \frac{x}{\sqrt{3}} + 30 \tag{1} \]

### Step 4: Use Triangle from Point A
Now, consider the triangle formed from point A to the top of the tower:
\[ \tan(45^\circ) = \frac{PQ}{x} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{PQ}{x} \implies PQ = x \tag{2} \]

### Step 5: Set Equations Equal
Now we have two expressions for \( PQ \):
From (1): \( PQ = \frac{x}{\sqrt{3}} + 30 \)
From (2): \( PQ = x \)

Setting them equal gives:
\[ x = \frac{x}{\sqrt{3}} + 30 \]

### Step 6: Solve for x
Rearranging the equation:
\[ x - \frac{x}{\sqrt{3}} = 30 \]
Factoring out \( x \):
\[ x \left(1 - \frac{1}{\sqrt{3}}\right) = 30 \]
To simplify \( 1 - \frac{1}{\sqrt{3}} \):
\[ 1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}} \]
Thus, we have:
\[ x \cdot \frac{\sqrt{3} - 1}{\sqrt{3}} = 30 \]
Multiplying both sides by \( \frac{\sqrt{3}}{\sqrt{3} - 1} \):
\[ x = 30 \cdot \frac{\sqrt{3}}{\sqrt{3} - 1} \]

### Step 7: Rationalize the Denominator
To rationalize the denominator:
\[ x = 30 \cdot \frac{\sqrt{3}(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = 30 \cdot \frac{\sqrt{3}(\sqrt{3} + 1)}{2} \]
This simplifies to:
\[ x = 15\sqrt{3}(\sqrt{3} + 1) = 15(3 + \sqrt{3}) = 45 + 15\sqrt{3} \]

### Final Answer
The distance of the foot of the tower from point A is \( 15(3 + \sqrt{3}) \) meters.

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