The angle of elevation of the top of a hill from the foot of a tower is `60^(@)` and the angle of elevation of the top of the tower from the foot of the hill is `30^(@)`. If the tower is 50 m high, then what is the height of the hill?

Options

A. 180 m
B. 150 m
C. 100 m
D. 120 m

Explanation

To find the height of the hill, we can follow these steps:

### Step 1: Draw the diagram
Draw a diagram with a tower (CD) and a hill (AB). Mark the height of the tower (CD) as 50 m. Label the foot of the tower as point C and the foot of the hill as point A. The top of the tower is point D and the top of the hill is point B.

### Step 2: Identify the angles
From the foot of the tower (C), the angle of elevation to the top of the hill (B) is 60 degrees. From the foot of the hill (A), the angle of elevation to the top of the tower (D) is 30 degrees.

### Step 3: Use the triangle BDC to find BD
In triangle BDC, we have:
- Angle CDB = 60 degrees
- CD (height of the tower) = 50 m

Using the tangent function:
\[ \tan(60^\circ) = \frac{BD}{CD} \]
Substituting the known values:
\[ \sqrt{3} = \frac{BD}{50} \]
Now, solve for BD:
\[ BD = 50 \sqrt{3} \text{ m} \]

### Step 4: Use the triangle ABD to find AB
In triangle ABD, we have:
- Angle ADB = 30 degrees
- BD (base) = 50√3 m

Using the tangent function again:
\[ \tan(30^\circ) = \frac{AB}{BD} \]
Substituting the known values:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{50 \sqrt{3}} \]
Now, solve for AB:
\[ AB = 50 \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 50 \text{ m} \]

### Step 5: Calculate the total height of the hill
The total height of the hill (AB) is 50 m.

### Final Answer
The height of the hill is **150 m**.