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

Options

A. `(h sin alpha sin (alpha - beta))/(sin beta)`
B. `(h sin alpha cos (alpha + beta))/(cos beta)`
C. `(h sin alpha cos (alpha - beta))/(sin beta)`
D. `(h sin alpha sin (alpha +beta))/(cos beta)`

Explanation

To find the height of the tower (PQ) given the height of the pole (h), the angle of elevation to the top of the tower (α), and the angle subtended by the pole at the top of the tower (β), we can follow these steps:

### Step 1: Understand the Geometry
- Let the height of the pole be \( h \).
- Let the height of the tower be \( PQ \).
- The distance from the pole to the base of the tower is \( x \).
- The angle of elevation to the top of the tower from the bottom of the pole is \( \alpha \).
- The angle subtended by the pole at the top of the tower is \( \beta \).

### Step 2: Set Up the First Triangle (OPQ)
In triangle OPQ:
- We can use the tangent function to express the height of the tower in terms of \( x \):

\[ \tan(\alpha) = \frac{PQ}{x} \]

This implies:

\[ PQ = x \tan(\alpha) \]

### Step 3: Set Up the Second Triangle (ARQ)
In triangle ARQ:
- The angle at point R (the top of the tower) is \( \alpha - \beta \).
- We can again use the tangent function:

\[ \tan(\alpha - \beta) = \frac{QR}{AR} \]

Since \( AR = x \) (the distance from the pole to the tower), we can write:

\[ \tan(\alpha - \beta) = \frac{QR}{x} \]

This implies:

\[ QR = x \tan(\alpha - \beta) \]

### Step 4: Relate QR to the Height of the Pole
Since \( QR = PQ - h \), we can substitute:

\[ PQ - h = x \tan(\alpha - \beta) \]

### Step 5: Substitute for PQ
From our earlier equation for \( PQ \):

\[ x \tan(\alpha) - h = x \tan(\alpha - \beta) \]

### Step 6: Rearranging the Equation
Rearranging gives:

\[ x \tan(\alpha) - x \tan(\alpha - \beta) = h \]

Factoring out \( x \):

\[ x (\tan(\alpha) - \tan(\alpha - \beta)) = h \]

### Step 7: Solve for x
Thus, we can express \( x \) as:

\[ x = \frac{h}{\tan(\alpha) - \tan(\alpha - \beta)} \]

### Step 8: Substitute x Back into PQ
Now, substitute \( x \) back into the equation for \( PQ \):

\[ PQ = x \tan(\alpha) = \left(\frac{h}{\tan(\alpha) - \tan(\alpha - \beta)}\right) \tan(\alpha) \]

### Step 9: Final Expression for PQ
This gives us:

\[ PQ = \frac{h \tan(\alpha)}{\tan(\alpha) - \tan(\alpha - \beta)} \]

### Step 10: Simplifying Using Trigonometric Identities
Using the identity for the tangent of a difference:

\[ \tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha) \tan(\beta)} \]

We can rewrite the final expression for \( PQ \) if needed.

### Final Answer
The height of the tower \( PQ \) is given by:

\[ PQ = \frac{h \sin(\alpha) \cos(\alpha - \beta)}{\sin(\beta)} \]