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

Options

A. `(h cot q)/(cot q - cot p)`
B. `(h cot p)/(cot p-cot q)`
C. `(h tan p)/( tan p - tan q)`
D. none of these

Answer

To find the height of the hill (denoted as \( H \)), we can use the information provided about the angles of elevation from the top and bottom of a building of height \( h \). Let's break down the solution step by step.

### Step 1: Understand the Geometry
- Let the height of the hill be \( H \).
- The height of the building is \( h \).
- The angle of elevation from the top of the building to the top of the hill is \( p \).
- The angle of elevation from the bottom of the building to the top of the hill is \( q \).

### Step 2: Set Up the Triangles
1. From the top of the building (point A) to the top of the hill (point B), we can form a right triangle:
- The height of the hill above the top of the building is \( H - h \).
- The horizontal distance from the building to the hill is \( x \).
- Therefore, we can write the equation using the tangent function:
\[ \tan(p) = \frac{H - h}{x} \]
Rearranging gives:
\[ H - h = x \tan(p) \quad \text{(1)} \]

2. From the bottom of the building (point C) to the top of the hill (point B), we form another right triangle:
- The height of the hill is \( H \).
- The horizontal distance remains the same \( x \).
- Thus, we have:
\[ \tan(q) = \frac{H}{x} \]
Rearranging gives:
\[ H = x \tan(q) \quad \text{(2)} \]

### Step 3: Solve for \( x \)
From equation (1), we can express \( x \) in terms of \( H \):
\[ x = \frac{H - h}{\tan(p)} \quad \text{(3)} \]

### Step 4: Substitute \( x \) into Equation (2)
Now substitute equation (3) into equation (2):
\[ H = \left(\frac{H - h}{\tan(p)}\right) \tan(q) \]
Cross-multiplying gives:
\[ H \tan(p) = (H - h) \tan(q) \]

### Step 5: Rearrange the Equation
Expanding and rearranging:
\[ H \tan(p) = H \tan(q) - h \tan(q) \]
\[ H \tan(p) - H \tan(q) = -h \tan(q) \]
\[ H (\tan(p) - \tan(q)) = h \tan(q) \]

### Step 6: Solve for \( H \)
Finally, we can solve for \( H \):
\[ H = \frac{h \tan(q)}{\tan(p) - \tan(q)} \]

### Conclusion
Thus, the height of the hill \( H \) is given by:
\[ H = h \frac{\tan(q)}{\tan(p) - \tan(q)} \]