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 use trigonometric concepts and properties of right triangles. Let's break down the solution step by step.

### Step 1: Understand the Problem
We have a tower of height \( H \) meters. At a point on the same level as the foot of the tower, the angle of elevation to the top of the tower is \( 75^\circ \). At another point, which is \( 10 \) meters above the first point, the angle of depression to the foot of the tower is \( 15^\circ \).

### Step 2: Set Up the Diagram
1. Let \( A \) be the top of the tower.
2. Let \( B \) be the foot of the tower.
3. Let \( C \) be the point on the same level as \( B \) where the angle of elevation is measured.
4. Let \( D \) be the point \( 10 \) meters above \( C \).

### Step 3: Define the Variables
- Let \( BC = x \) (the horizontal distance from point \( C \) to the foot of the tower).
- The height of the tower \( AB = H \).

### Step 4: Use Trigonometric Ratios
1. From point \( C \) (where angle of elevation is \( 75^\circ \)):
\[ \tan(75^\circ) = \frac{H}{x} \]
Therefore,
\[ H = x \cdot \tan(75^\circ) \]

2. From point \( D \) (10 meters above \( C \), where angle of depression is \( 15^\circ \)):
The height from point \( D \) to the foot of the tower \( B \) is \( H - 10 \). The angle of depression to \( B \) means:
\[ \tan(15^\circ) = \frac{H - 10}{x} \]
Therefore,
\[ H - 10 = x \cdot \tan(15^\circ) \]

### Step 5: Set Up the Equations
Now we have two equations:
1. \( H = x \cdot \tan(75^\circ) \) (1)
2. \( H - 10 = x \cdot \tan(15^\circ) \) (2)

### Step 6: Substitute and Solve
From equation (1), we can express \( x \):
\[ x = \frac{H}{\tan(75^\circ)} \]

Substituting \( x \) into equation (2):
\[ H - 10 = \left(\frac{H}{\tan(75^\circ)}\right) \cdot \tan(15^\circ) \]

### Step 7: Rearranging the Equation
Multiply both sides by \( \tan(75^\circ) \):
\[ (H - 10) \cdot \tan(75^\circ) = H \cdot \tan(15^\circ) \]

Expanding gives:
\[ H \cdot \tan(75^\circ) - 10 \cdot \tan(75^\circ) = H \cdot \tan(15^\circ) \]

### Step 8: Isolate \( H \)
Rearranging gives:
\[ H \cdot (\tan(75^\circ) - \tan(15^\circ)) = 10 \cdot \tan(75^\circ) \]
Thus,
\[ H = \frac{10 \cdot \tan(75^\circ)}{\tan(75^\circ) - \tan(15^\circ)} \]

### Step 9: Calculate Values
Using known values:
- \( \tan(75^\circ) \approx 3.732 \)
- \( \tan(15^\circ) \approx 0.2679 \)

Substituting these values:
\[ H = \frac{10 \cdot 3.732}{3.732 - 0.2679} \approx \frac{37.32}{3.4641} \approx 10.77 \text{ meters} \]

### Final Answer
The height of the tower is approximately \( 10.77 \) meters.

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