From the top of a cliff 300 metres high, the top of a tower was observed at an angle of depression `30^@` and from the foot of the tower the top of the cliff was observed at an angle of elevation `45^@`, The height of the tower is

Options

A. `50(3-sqrt3)m`
B. `200(3-sqrt3)` m
C. `100(3-sqrt3)m`
D. none of these

Answer

To solve the problem step by step, we will use trigonometric ratios and the information given in the question.

### Step 1: Understand the Problem and Draw a Diagram
We have a cliff of height 300 meters and a tower. From the top of the cliff, the angle of depression to the top of the tower is 30 degrees. From the foot of the tower, the angle of elevation to the top of the cliff is 45 degrees. We need to find the height of the tower.

### Step 2: Label the Diagram
Let's label the points:
- A: Top of the cliff
- B: Foot of the cliff
- C: Top of the tower
- D: Foot of the tower

Given:
- Height of the cliff (AB) = 300 meters
- Angle of depression from A to C = 30 degrees
- Angle of elevation from D to A = 45 degrees

### Step 3: Use the Angle of Elevation to Find Distance BD
From point D (foot of the tower), we can use the angle of elevation to find the horizontal distance (BD) from the foot of the cliff to the foot of the tower.

Using the tangent function:
\[ \tan(45^\circ) = \frac{AB}{BD} \]
Since \(\tan(45^\circ) = 1\):
\[ 1 = \frac{300}{BD} \implies BD = 300 \text{ meters} \]

### Step 4: Use the Angle of Depression to Find Height of Tower (DC)
Now, we will use the angle of depression from A to C (the top of the tower). The angle of depression of 30 degrees corresponds to the angle of elevation from C to A.

Using the tangent function:
\[ \tan(30^\circ) = \frac{AB - DC}{BD} \]
Where \(DC\) is the height of the tower. We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{300 - DC}{300} \]

### Step 5: Solve for DC
Cross-multiplying gives:
\[ 300 = (300 - DC) \sqrt{3} \]
Expanding:
\[ 300 = 300\sqrt{3} - DC\sqrt{3} \]
Rearranging gives:
\[ DC\sqrt{3} = 300\sqrt{3} - 300 \]
\[ DC = \frac{300(\sqrt{3} - 1)}{\sqrt{3}} \]

### Step 6: Simplify DC
To simplify, we can multiply the numerator and denominator by \(\sqrt{3}\):
\[ DC = 300 \left(1 - \frac{1}{\sqrt{3}}\right) = 300 \left(1 - \frac{\sqrt{3}}{3}\right) = 300 \left(\frac{3 - \sqrt{3}}{3}\right) = 100(3 - \sqrt{3}) \]

### Final Answer
The height of the tower is:
\[ \text{Height of the tower (DC)} = 100(3 - \sqrt{3}) \text{ meters} \]