The angle of elevation of the top of a tower standing on a horizontal plane from a point A is `alpha`. After walking a distance d towards the foot of the tower, the angle of elevation is found to be `beta`. The height of the tower is

Options

A. `(dsinalphasinbeta)/(sin(beta-alpha))`
B. `(dsinalphasinbeta)/(sin(alpha-beta))`
C. `(dsin(beta-alpha))/(sinalphasinbeta)`
D. `(dsin(alpha-beta))/(sinalphasinbeta)`

Answer

To solve the problem of finding the height of the tower given the angles of elevation from two different points, we can follow these steps:

### Step 1: Draw the Diagram
Draw a vertical line to represent the tower (let's call it point C). Mark point A as the initial observation point from which the angle of elevation to the top of the tower is α. After walking a distance d towards the tower, mark point B as the new observation point, where the angle of elevation is β.

### Step 2: Identify the Right Triangles
From point A, we can form a right triangle ACB where:
- AC is the height of the tower (h).
- AB is the horizontal distance from point A to the base of the tower (let's denote it as x).
- The angle of elevation from A to C is α.

From point B, we can form another right triangle BCD where:
- BC is the height of the tower (h).
- BD is the horizontal distance from point B to the base of the tower, which is (x - d).
- The angle of elevation from B to C is β.

### Step 3: Write the Tangent Equations
Using the tangent function for both triangles, we can write:
1. From triangle ACB:
\[ \tan(\alpha) = \frac{h}{x} \]
Rearranging gives:
\[ h = x \tan(\alpha) \quad (1) \]

2. From triangle BCD:
\[ \tan(\beta) = \frac{h}{x - d} \]
Rearranging gives:
\[ h = (x - d) \tan(\beta) \quad (2) \]

### Step 4: Set the Equations Equal
Since both equations equal h, we can set them equal to each other:
\[ x \tan(\alpha) = (x - d) \tan(\beta) \]

### Step 5: Solve for x
Expanding the right side gives:
\[ x \tan(\alpha) = x \tan(\beta) - d \tan(\beta) \]
Rearranging terms gives:
\[ x \tan(\alpha) - x \tan(\beta) = -d \tan(\beta) \]
Factoring out x:
\[ x (\tan(\alpha) - \tan(\beta)) = -d \tan(\beta) \]
Thus, we can solve for x:
\[ x = \frac{-d \tan(\beta)}{\tan(\alpha) - \tan(\beta)} \quad (3) \]

### Step 6: Substitute x Back to Find h
Now substitute equation (3) back into either equation (1) or (2) to find h. Using equation (1):
\[ h = \left(\frac{-d \tan(\beta)}{\tan(\alpha) - \tan(\beta)}\right) \tan(\alpha) \]
Simplifying gives:
\[ h = \frac{-d \tan(\beta) \tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]

### Final Expression for h
Thus, the height of the tower h can be expressed as:
\[ h = \frac{d \tan(\alpha) \tan(\beta)}{\tan(\beta) - \tan(\alpha)} \]