From the top of a pole of height h, the angle of elevation of the top of the tower is `alpha`. The pole subtends an angle `beta` at the top of the tower. The height of the tower is____

Answer

To find the height of the tower, we can follow these steps:

### Step 1: Understand the Problem
We have a pole of height \( h \) and a tower. From the top of the pole, the angle of elevation to the top of the tower is \( \alpha \), and the angle subtended by the pole at the top of the tower is \( \beta \).

### Step 2: Set Up the Diagram
Draw a diagram with:
- A vertical pole of height \( h \).
- A tower next to the pole.
- Label the top of the pole as point \( P \) and the top of the tower as point \( Q \).
- Let \( R \) be the point where the pole meets the ground.

### Step 3: Identify the Angles
From the top of the pole \( P \):
- The angle of elevation to the top of the tower \( Q \) is \( \alpha \).
- The angle subtended by the pole at the top of the tower is \( \beta \).

### Step 4: Use Right Triangle Trigonometry
1. **For Triangle \( OPQ \)**:
- The height \( PQ \) can be expressed using the tangent function:
\[ PQ = OP \cdot \tan(\alpha) \]
- Let \( OP \) be the horizontal distance from the pole to the base of the tower, denoted as \( x \). Thus:
\[ PQ = x \cdot \tan(\alpha) \]

2. **For Triangle \( ARQ \)**:
- The height \( QR \) (the height of the tower above the pole) can be expressed using the angle \( \beta \):
\[ QR = OP \cdot \tan(\beta) \]
- Therefore:
\[ QR = x \cdot \tan(\beta) \]

### Step 5: Relate the Heights
The total height of the tower \( h_T \) can be expressed as:
\[ h_T = h + QR \]
Substituting for \( QR \):
\[ h_T = h + x \cdot \tan(\beta) \]

### Step 6: Substitute \( x \) from the First Triangle
From the first triangle, we have:
\[ x = \frac{PQ}{\tan(\alpha)} \]
Substituting this into the equation for \( h_T \):
\[ h_T = h + \left(\frac{PQ}{\tan(\alpha)}\right) \cdot \tan(\beta) \]

### Step 7: Solve for \( PQ \)
We can express \( PQ \) in terms of \( h \), \( \alpha \), and \( \beta \):
\[ PQ = h_T - h \]
Substituting back, we find:
\[ PQ = h + \left(\frac{PQ}{\tan(\alpha)}\right) \cdot \tan(\beta) - h \]

### Step 8: Final Expression for the Height of the Tower
After simplifying, we arrive at the final expression for the height of the tower:
\[ h_T = \frac{h \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\alpha) - \tan(\beta)} \]

### Final Answer
The height of the tower is:
\[ \text{Height of the tower} = \frac{h \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\alpha) - \tan(\beta)} \]