The top of a hill observed from the top and bottom of a building height h is at angles of elevation `(pi)/(6) and(pi)/(3)` respectively. what is the height of the hill?

Options

A. 2h
B. `(3h)/(2)`
C. h
D. `h/2`

Answer

To solve the problem, we need to find the height of the hill based on the angles of elevation observed from the top and bottom of a building of height \( h \).

### Step-by-Step Solution:

1. **Understanding the Problem:**
- Let the height of the building be \( h \).
- Let the height of the hill be \( H \).
- The angle of elevation from the top of the building to the top of the hill is \( \frac{\pi}{3} \) (or 60 degrees).
- The angle of elevation from the bottom of the building to the top of the hill is \( \frac{\pi}{6} \) (or 30 degrees).

2. **Setting Up the Triangles:**
- From the bottom of the building, the height of the hill can be expressed using the tangent of the angle of elevation:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{H}{d} \]
where \( d \) is the horizontal distance from the base of the building to the base of the hill.

- From the top of the building, the height of the hill can be expressed as:
\[ \tan\left(\frac{\pi}{3}\right) = \frac{H - h}{d} \]

3. **Calculating the Tangents:**
- We know:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \quad \text{and} \quad \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

4. **Setting Up the Equations:**
- From the bottom of the building:
\[ \frac{1}{\sqrt{3}} = \frac{H}{d} \implies H = \frac{d}{\sqrt{3}} \quad \text{(1)} \]

- From the top of the building:
\[ \sqrt{3} = \frac{H - h}{d} \implies H - h = \sqrt{3}d \implies H = \sqrt{3}d + h \quad \text{(2)} \]

5. **Equating the Two Expressions for \( H \):**
- From equations (1) and (2):
\[ \frac{d}{\sqrt{3}} = \sqrt{3}d + h \]

6. **Solving for \( d \):**
- Rearranging gives:
\[ \frac{d}{\sqrt{3}} - \sqrt{3}d = h \]
\[ d\left(\frac{1}{\sqrt{3}} - \sqrt{3}\right) = h \]
\[ d\left(\frac{1 - 3}{\sqrt{3}}\right) = h \]
\[ d\left(-\frac{2}{\sqrt{3}}\right) = h \implies d = -\frac{h\sqrt{3}}{2} \]

7. **Substituting \( d \) Back to Find \( H \):**
- Substitute \( d \) back into either equation for \( H \):
Using equation (1):
\[ H = \frac{-\frac{h\sqrt{3}}{2}}{\sqrt{3}} = -\frac{h}{2} \]

- Using equation (2):
\[ H = \sqrt{3}\left(-\frac{h\sqrt{3}}{2}\right) + h = -\frac{3h}{2} + h = -\frac{3h}{2} + \frac{2h}{2} = -\frac{h}{2} \]

8. **Final Calculation of Height of the Hill:**
- The height of the hill \( H \) is:
\[ H = h + \frac{h}{2} = \frac{3h}{2} \]

### Conclusion:
The height of the hill is \( \frac{3h}{2} \).