The top of a hill observed from the top and bottom of a building of height h is at angles of elevation `alpha and beta.` respectively. The height of the bill is

Options

A. `(h cot beta)/( cot beta - cot alpha )`
B. `( h cot alpha )/( cot alpha - cot beta)`
C. `( h cot alpha )/( tan alpha - tan beta)`
D. None of these

Answer

To find the height of the hill (let's denote it as \( BE \)), we can follow these steps:

### Step 1: Define the Variables
- Let \( h \) be the height of the building.
- Let \( x \) be the height of the hill above the building.
- Therefore, the total height of the hill \( BE = h + x \).
- Let \( Y \) be the horizontal distance from the building to the base of the hill.

### Step 2: Set Up the First Triangle (from the top of the building)
From the top of the building (point A), the angle of elevation to the top of the hill (point E) is \( \alpha \). We can use the tangent function:
\[ \tan(\alpha) = \frac{BE}{Y} = \frac{h + x}{Y} \]
This can be rearranged to:
\[ Y = \frac{h + x}{\tan(\alpha)} \]

### Step 3: Set Up the Second Triangle (from the bottom of the building)
From the bottom of the building (point B), the angle of elevation to the top of the hill (point E) is \( \beta \). Again, using the tangent function:
\[ \tan(\beta) = \frac{BE}{Y} = \frac{h + x}{Y} \]
This can be rearranged to:
\[ Y = \frac{h + x}{\tan(\beta)} \]

### Step 4: Equate the Two Expressions for Y
Since both expressions represent the same horizontal distance \( Y \), we can set them equal to each other:
\[ \frac{h + x}{\tan(\alpha)} = \frac{h + x}{\tan(\beta)} \]

### Step 5: Solve for x
Cross-multiplying gives:
\[ (h + x) \tan(\beta) = (h + x) \tan(\alpha) \]
Assuming \( h + x \neq 0 \), we can divide both sides by \( (h + x) \):
\[ \tan(\beta) = \tan(\alpha) \]

### Step 6: Rearranging for x
From the two equations for \( Y \), we can express \( h + x \) in terms of \( Y \):
\[ h + x = Y \tan(\alpha) \]
Substituting \( Y \) from the second triangle gives:
\[ h + x = \frac{h + x}{\tan(\beta)} \tan(\alpha) \]
Rearranging gives:
\[ h + x = h \frac{\tan(\alpha)}{\tan(\beta)} + x \frac{\tan(\alpha)}{\tan(\beta)} \]

### Step 7: Isolate x
Rearranging the equation to isolate \( x \):
\[ x \left(1 - \frac{\tan(\alpha)}{\tan(\beta)}\right) = h \frac{\tan(\alpha)}{\tan(\beta)} - h \]
Solving for \( x \):
\[ x = \frac{h \left(\tan(\alpha) - \tan(\beta)\right)}{\tan(\beta) - \tan(\alpha)} \]

### Step 8: Find the Total Height of the Hill
Now, substituting \( x \) back into \( BE \):
\[ BE = h + x \]
Thus:
\[ BE = h + \frac{h \left(\tan(\alpha) - \tan(\beta)\right)}{\tan(\beta) - \tan(\alpha)} \]
This simplifies to:
\[ BE = h \left(1 + \frac{\tan(\alpha) - \tan(\beta)}{\tan(\beta) - \tan(\alpha)}\right) \]

### Final Result
The height of the hill \( BE \) can be expressed as:
\[ BE = h \cdot \frac{\tan(\alpha)}{\tan(\alpha) - \tan(\beta)} \]