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 - cotq)`
C. `(h tan p)/(tan p - tanq)`
D. none of these

Answer

To solve the problem of finding the height of the hill (denoted as \( x \)), we will use trigonometric relationships based on the angles of elevation from the top and bottom of a building of height \( h \).

### Step-by-Step Solution:

1. **Draw the Diagram**:
- Let the height of the hill be \( x \).
- Let the height of the building be \( h \).
- From the top of the building, the angle of elevation to the top of the hill is \( p \).
- From the bottom of the building, the angle of elevation to the top of the hill is \( q \).

2. **Identify Points**:
- Let point \( A \) be the top of the hill.
- Let point \( B \) be the top of the building.
- Let point \( C \) be the bottom of the building.
- Let point \( D \) be the point directly below the top of the hill at the same horizontal level as point \( C \).
- Let point \( E \) be the point where the angle of elevation \( q \) is measured from point \( C \).

3. **Using Triangle Relationships**:
- In triangle \( ABE \) (formed by the top of the hill, top of the building, and the horizontal line):
\[ \tan(p) = \frac{AB}{BE} \]
Here, \( AB = x - h \) (the height from the top of the building to the top of the hill) and \( BE = d \) (the horizontal distance).
Thus, we have:
\[ \tan(p) = \frac{x - h}{d} \quad \text{(1)} \]

- In triangle \( ACD \) (formed by the top of the hill, bottom of the building, and the horizontal line):
\[ \tan(q) = \frac{AC}{CD} \]
Here, \( AC = x \) (the height of the hill) and \( CD = d \).
Thus, we have:
\[ \tan(q) = \frac{x}{d} \quad \text{(2)} \]

4. **Express \( d \) in Terms of \( x \)**:
- From equation (2), we can express \( d \):
\[ d = \frac{x}{\tan(q)} \quad \text{(3)} \]

5. **Substituting \( d \) into Equation (1)**:
- Substitute equation (3) into equation (1):
\[ \tan(p) = \frac{x - h}{\frac{x}{\tan(q)}} \]
- This simplifies to:
\[ \tan(p) = \frac{(x - h) \tan(q)}{x} \]

6. **Cross-Multiplying**:
- Cross-multiply to eliminate the fraction:
\[ x \tan(p) = (x - h) \tan(q) \]

7. **Expanding and Rearranging**:
- Expand the right side:
\[ x \tan(p) = x \tan(q) - h \tan(q) \]
- Rearranging gives:
\[ x \tan(p) - x \tan(q) = -h \tan(q) \]
- Factor out \( x \):
\[ x (\tan(p) - \tan(q)) = -h \tan(q) \]

8. **Solving for \( x \)**:
- Finally, solving for \( x \):
\[ x = \frac{h \tan(q)}{\tan(p) - \tan(q)} \]

### Final Result:
The height of the hill \( x \) is given by:
\[ x = \frac{h \tan(q)}{\tan(p) - \tan(q)} \]