The top of a hill observed from the top and bottom of a building of 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=3h
D. `(h)/(2)`

Explanation

To find the height of the hill, let's denote the height of the hill as \( H \) and the height of the building as \( h \). The angles of elevation from the top and bottom of the building to the top of the hill are given as \( \frac{\pi}{6} \) and \( \frac{\pi}{3} \), respectively.

### Step 1: Set Up the Problem
From the bottom of the building, the angle of elevation to the top of the hill is \( \frac{\pi}{6} \). From the top of the building, the angle of elevation to the top of the hill is \( \frac{\pi}{3} \).

### Step 2: Use Trigonometric Ratios
Using the tangent function, we can write the following equations based on the right triangles formed:

1. From the bottom of the building:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{H - h}{d} \]
where \( d \) is the horizontal distance from the base of the building to the hill.

2. From the top of the building:
\[ \tan\left(\frac{\pi}{3}\right) = \frac{H}{d} \]

### Step 3: Calculate the Tangent Values
We know:
- \( \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \)
- \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \)

Substituting these values into the equations gives us:

1. From the bottom:
\[ \frac{1}{\sqrt{3}} = \frac{H - h}{d} \quad \Rightarrow \quad H - h = \frac{d}{\sqrt{3}} \quad \text{(1)} \]

2. From the top:
\[ \sqrt{3} = \frac{H}{d} \quad \Rightarrow \quad H = d \sqrt{3} \quad \text{(2)} \]

### Step 4: Substitute Equation (2) into Equation (1)
Now, substitute \( H \) from equation (2) into equation (1):

\[ d \sqrt{3} - h = \frac{d}{\sqrt{3}} \]

### Step 5: Solve for \( d \)
Rearranging gives us:

\[ d \sqrt{3} - \frac{d}{\sqrt{3}} = h \]

Factoring out \( d \):

\[ d\left(\sqrt{3} - \frac{1}{\sqrt{3}}\right) = h \]

To simplify \( \sqrt{3} - \frac{1}{\sqrt{3}} \):

\[ \sqrt{3} - \frac{1}{\sqrt{3}} = \frac{3 - 1}{\sqrt{3}} = \frac{2}{\sqrt{3}} \]

Thus, we have:

\[ d \cdot \frac{2}{\sqrt{3}} = h \quad \Rightarrow \quad d = \frac{h \sqrt{3}}{2} \quad \text{(3)} \]

### Step 6: Substitute \( d \) Back to Find \( H \)
Now substitute \( d \) from equation (3) back into equation (2):

\[ H = d \sqrt{3} = \left(\frac{h \sqrt{3}}{2}\right) \sqrt{3} = \frac{h \cdot 3}{2} = \frac{3h}{2} \]

### Final Answer
Thus, the height of the hill is:

\[ H = \frac{3h}{2} \]