From the top of a cliff 50 m high, the angles of depression of the top and bottom of a tower are observed to be `30^(@)` and `45^(@).` The height of tower is

Options

A. `50 m`
B. `50 sqrt3 m`
C. `50 (sqrt3 -1) m`
D. `50 (1- (sqrt3)/(3)) m`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a cliff that is 50 m high. From the top of the cliff, the angles of depression to the top and bottom of a tower are given as 30° and 45°, respectively. We need to find the height of the tower.

### Step 2: Draw a Diagram
Draw a diagram to visualize the situation:
- Let point A be the top of the cliff.
- Let point B be the bottom of the tower.
- Let point C be the top of the tower.
- The height of the cliff (AB) = 50 m.
- The angle of depression to point C (top of the tower) = 30°.
- The angle of depression to point B (bottom of the tower) = 45°.

### Step 3: Establish Relationships
From the angles of depression, we can establish relationships using trigonometric functions:
- The angle of depression to point C (top of the tower) creates a right triangle with the height of the cliff and the horizontal distance from the cliff to the tower.
- The angle of depression to point B (bottom of the tower) also creates a right triangle.

### Step 4: Set Up the Equations
Let:
- \( h \) = height of the tower (BC)
- \( d \) = horizontal distance from the cliff to the tower (AD)

From the triangle formed by the angle of depression to the top of the tower (30°):
\[ \tan(30°) = \frac{h + 50}{d} \]
Using \( \tan(30°) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h + 50}{d} \quad \text{(1)} \]

From the triangle formed by the angle of depression to the bottom of the tower (45°):
\[ \tan(45°) = \frac{50}{d} \]
Using \( \tan(45°) = 1 \):
\[ 1 = \frac{50}{d} \quad \text{(2)} \]

### Step 5: Solve for \( d \)
From equation (2):
\[ d = 50 \]

### Step 6: Substitute \( d \) into Equation (1)
Now substitute \( d = 50 \) into equation (1):
\[ \frac{1}{\sqrt{3}} = \frac{h + 50}{50} \]

### Step 7: Solve for \( h \)
Cross-multiply to solve for \( h \):
\[ 50 = (h + 50) \cdot \frac{1}{\sqrt{3}} \]
\[ 50\sqrt{3} = h + 50 \]
\[ h = 50\sqrt{3} - 50 \]

### Step 8: Final Calculation
To find the height of the tower, we can factor out 50:
\[ h = 50(\sqrt{3} - 1) \]

### Conclusion
The height of the tower is:
\[ h = 50(\sqrt{3} - 1) \text{ meters} \]