A tower subtends an angle `75^(@)` at a point on the same level as the foot of the tower and at another point, 10 meters above the first, the angle of depression of the foot of the tower is `15^(@)`. The height of the tower is (in meters)

Options

A. `10(sqrt3+1)^(2)`
B. `10(sqrt3-1)^(2)`
C. `10(2+sqrt3)^(2)`
D. None of these

Answer

To solve the problem, we will analyze the given information step by step.

### Step 1: Understand the Geometry
Let's denote:
- \( AB \) as the height of the tower.
- \( C \) as the point on the same level as the foot of the tower where the angle of elevation is measured.
- \( D \) as the point 10 meters above point \( C \).

From point \( C \), the angle of elevation to the top of the tower \( A \) is \( 75^\circ \). From point \( D \), the angle of depression to the foot of the tower \( B \) is \( 15^\circ \).

### Step 2: Set Up the Right Triangles
1. From point \( C \):
- The angle of elevation to the top of the tower \( A \) is \( 75^\circ \).
- Let the distance from point \( C \) to the foot of the tower \( B \) be \( x \).
- Using the tangent function:
\[ \tan(75^\circ) = \frac{AB}{x} \]
- Therefore, we can express the height of the tower \( AB \) as:
\[ AB = x \cdot \tan(75^\circ) \]

2. From point \( D \):
- The angle of depression to the foot of the tower \( B \) is \( 15^\circ \).
- The height from point \( D \) to point \( C \) is 10 meters.
- The distance from point \( D \) to the foot of the tower \( B \) is still \( x \).
- Using the tangent function:
\[ \tan(15^\circ) = \frac{10}{x} \]
- Rearranging gives:
\[ x = \frac{10}{\tan(15^\circ)} \]

### Step 3: Substitute and Solve
Now we can substitute \( x \) from the second equation into the first equation:
\[ AB = \left(\frac{10}{\tan(15^\circ)}\right) \cdot \tan(75^\circ) \]

### Step 4: Calculate the Values
Using the known values of the tangent functions:
- \( \tan(75^\circ) = 2 + \sqrt{3} \)
- \( \tan(15^\circ) = 2 - \sqrt{3} \)

Substituting these values:
\[ AB = \left(\frac{10}{2 - \sqrt{3}}\right) \cdot (2 + \sqrt{3}) \]

### Step 5: Rationalize the Denominator
To simplify:
\[ AB = 10 \cdot \frac{(2 + \sqrt{3})}{(2 - \sqrt{3})} \cdot \frac{(2 + \sqrt{3})}{(2 + \sqrt{3})} \]
\[ = 10 \cdot \frac{(2 + \sqrt{3})^2}{(2^2 - (\sqrt{3})^2)} = 10 \cdot \frac{(4 + 4\sqrt{3} + 3)}{(4 - 3)} = 10 \cdot (7 + 4\sqrt{3}) \]
\[ = 70 + 40\sqrt{3} \]

### Final Answer
Thus, the height of the tower \( AB \) is:
\[ AB = 10(2 + \sqrt{3}) \text{ meters} \]