From the top of a cliff 200 m high, the angles of depression of the top and bottom of a tower are observed to be `30^@ and 45^@`, respectivley. What is the height of to tower ?

Options

A. 400 m
B. `400sqrt3`m
C. `400/sqrt3`m
D. None of these

Answer

To find the height of the tower from the top of a cliff that is 200 m high, where the angles of depression to the top and bottom of the tower are 30° and 45° respectively, we can follow these steps:

### Step 1: Understand the Geometry
- Let the height of the tower be \( h \).
- The height of the cliff is given as 200 m.
- The angle of depression to the top of the tower is 30°, and to the bottom of the tower is 45°.

### Step 2: Draw the Diagram
- Draw a vertical line representing the cliff (200 m).
- From the top of the cliff, draw lines at angles of depression of 30° and 45° to represent the top and bottom of the tower.

### Step 3: Identify the Points
- Let point Q be the top of the cliff.
- Let point A be the top of the tower.
- Let point B be the bottom of the tower.
- The distance from point Q to point B (the bottom of the tower) is 200 m - \( h \).

### Step 4: Use Trigonometry for the Bottom of the Tower
- In triangle QBD (where D is the point directly below Q on the ground):
- The angle of depression to point B is 45°.
- Therefore, \( \tan(45°) = \frac{200}{DB} \) where DB is the horizontal distance from the base of the cliff to the base of the tower.
- Since \( \tan(45°) = 1 \), we have:
\[ 1 = \frac{200}{DB} \implies DB = 200 \text{ m} \]

### Step 5: Use Trigonometry for the Top of the Tower
- In triangle QAE:
- The angle of depression to point A is 30°.
- Therefore, \( \tan(30°) = \frac{200 - h}{DB} \).
- Since \( \tan(30°) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{200 - h}{200} \]

### Step 6: Solve for \( h \)
- Cross-multiplying gives:
\[ 200 = (200 - h) \sqrt{3} \]
- Expanding this:
\[ 200 = 200\sqrt{3} - h\sqrt{3} \]
- Rearranging gives:
\[ h\sqrt{3} = 200\sqrt{3} - 200 \]
- Dividing by \( \sqrt{3} \):
\[ h = \frac{200(\sqrt{3} - 1)}{\sqrt{3}} \]

### Step 7: Final Calculation
- This gives us the height of the tower \( h \).

### Summary of the Solution
The height of the tower is:
\[ h = \frac{200(\sqrt{3} - 1)}{\sqrt{3}} \text{ meters} \]