PQ is a post of height a, AB is a tower of height h at a distance x from the post, and `alpha and beta` are the angles of elevation of B, at P and Q respectively such that `alpha gt beta` . Then

Options

A. `h = x tan alpha`
B. `h = x sin 2alpha`
C. `h = xcos 2 alpha`
D. `h=x sin alpha`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a post PQ of height \( a \) and a tower AB of height \( h \). The distance between the base of the post and the tower is \( x \). The angles of elevation from points P and Q to point B (the top of the tower) are \( \alpha \) and \( \beta \) respectively, with \( \alpha > \beta \).

### Step 2: Draw the Diagram
Draw a diagram to visualize the problem:
- Draw a vertical line for the post PQ with height \( a \).
- Draw another vertical line for the tower AB with height \( h \).
- Mark the distance \( x \) between the base of the post (point Q) and the base of the tower (point A).
- Mark points P (the top of the post) and B (the top of the tower).
- Indicate angles \( \alpha \) (angle of elevation from Q to B) and \( \beta \) (angle of elevation from P to B).

### Step 3: Identify the Triangles
Identify the triangles formed:
- Triangle QAB for angle \( \alpha \) (from point Q to point B).
- Triangle PAB for angle \( \beta \) (from point P to point B).

### Step 4: Apply Trigonometric Ratios
For triangle QAB:
- The angle of elevation \( \alpha \) gives us the relation:
\[ \tan(\alpha) = \frac{h}{x} \]
Rearranging this gives:
\[ h = x \tan(\alpha) \]

### Step 5: Conclusion
Thus, the height of the tower \( h \) can be expressed as:
\[ h = x \tan(\alpha) \]

### Final Answer
The height of the tower \( h \) is given by the formula:
\[ h = x \tan(\alpha) \]