A tower subtends an angle `alpha` at a point A 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 the height of the tower is

Options

A. `b tan alpha.tanbeta`
B. `b(cot beta)/(cot alpha)`
C. `b(tan beta)/(tan alpha).sinalpha`
D. `bcot alpha.cotbeta`

Answer

To find the height of the tower given the angles α and β, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- Let the height of the tower be \( h \).
- Let point \( A \) be at the base of the tower, and point \( B \) be a point \( b \) feet above point \( A \).
- The angle \( \alpha \) is the angle subtended by the tower at point \( A \).
- The angle \( \beta \) is the angle of depression from point \( B \) to point \( A \).

2. **Identify the Right Triangles**:
- In triangle \( ABC \) (where \( C \) is the top of the tower), we can use the tangent function:
\[ \tan(\alpha) = \frac{h}{AB} \]
- Here, \( AB \) is the horizontal distance from point \( A \) to the base of the tower.

3. **Express \( AB \) in terms of \( h \)**:
- Rearranging the equation gives:
\[ AB = \frac{h}{\tan(\alpha)} \]

4. **Analyze Triangle \( ABD \)**:
- In triangle \( ABD \) (where \( D \) is directly below point \( B \)), we have:
\[ \tan(\beta) = \frac{h}{AB + b} \]
- Here, \( AB + b \) is the distance from point \( B \) to the foot of the tower.

5. **Express \( AB + b \) in terms of \( h \)**:
- Rearranging gives:
\[ AB + b = \frac{h}{\tan(\beta)} \]

6. **Substitute \( AB \) into the equation**:
- Substitute \( AB = \frac{h}{\tan(\alpha)} \) into the equation:
\[ \frac{h}{\tan(\alpha)} + b = \frac{h}{\tan(\beta)} \]

7. **Clear the fractions**:
- Multiply through by \( \tan(\alpha) \tan(\beta) \):
\[ h \tan(\beta) + b \tan(\alpha) \tan(\beta) = h \tan(\alpha) \]

8. **Rearrange to isolate \( h \)**:
- Rearranging gives:
\[ h \tan(\beta) - h \tan(\alpha) = -b \tan(\alpha) \tan(\beta) \]
- Factor out \( h \):
\[ h (\tan(\beta) - \tan(\alpha)) = -b \tan(\alpha) \tan(\beta) \]

9. **Solve for \( h \)**:
- Finally, we can express \( h \) as:
\[ h = \frac{b \tan(\alpha) \tan(\beta)}{\tan(\beta) - \tan(\alpha)} \]

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