A tower subtends an angle `alpha` at a point in the plane of its base and the angle of depression of the foot of the tower at a point b ft. just above A is `beta`. Then , height of the tower is

Options

A. `b tan alpha cot beta`
B. `b cot alpha tan beta`
C. `b tan alpha tan beta`
D. `b cot alpha cot beta`

Answer

To find the height of the tower based on the given angles, we can use trigonometric relationships. Let's denote the height of the tower as \( H \), the distance from the point where the angle \( \alpha \) is subtended to the base of the tower as \( AB \), and the height of the point \( B \) above point \( A \) as \( b \).

### Step-by-Step Solution:

1. **Understanding the Angles**:
- The angle \( \alpha \) is subtended at point \( A \) by the tower.
- The angle \( \beta \) is the angle of depression from point \( B \) to the foot of the tower.

2. **Setting Up the Right Triangle**:
- From point \( A \) to the top of the tower, we can form a right triangle where:
- The height of the tower is \( H \).
- The distance from point \( A \) to the base of the tower is \( AB \).
- According to trigonometric definitions, we have:
\[ \tan(\alpha) = \frac{H}{AB} \quad \text{(1)} \]

3. **Setting Up the Second Right Triangle**:
- From point \( B \) (which is at height \( b \) above point \( A \)) to the foot of the tower, we can form another right triangle where:
- The height from point \( B \) to the foot of the tower is \( b \).
- The angle of depression \( \beta \) gives us:
\[ \tan(\beta) = \frac{b}{AB} \quad \text{(2)} \]

4. **Expressing \( AB \) from Both Triangles**:
- From equation (1), we can express \( AB \):
\[ AB = \frac{H}{\tan(\alpha)} \quad \text{(3)} \]
- From equation (2), we can express \( AB \) as:
\[ AB = \frac{b}{\tan(\beta)} \quad \text{(4)} \]

5. **Equating the Two Expressions for \( AB \)**:
- Set equations (3) and (4) equal to each other:
\[ \frac{H}{\tan(\alpha)} = \frac{b}{\tan(\beta)} \]

6. **Solving for \( H \)**:
- Rearranging the equation gives:
\[ H = \frac{b \cdot \tan(\alpha)}{\tan(\beta)} \]

### Final Result:
The height of the tower \( H \) is given by:
\[ H = b \cdot \frac{\tan(\alpha)}{\tan(\beta)} \]