An observer on the top of a cliff, 200 m above the sea-level, observes the angles of depression of the two ships to be `45^(@) and 30^(@)` respectively. Find the distance between the ships, if the ships are
on the same side of the cliff,

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem and draw a diagram
We have a cliff that is 200 m high. From the top of the cliff, an observer sees two ships at angles of depression of 45° and 30°. We need to find the distance between the two ships.

### Step 2: Label the diagram
Let:
- Point P be the top of the cliff.
- Point Q be the sea level (200 m below P).
- Point A be the position of the first ship (at 45° angle of depression).
- Point B be the position of the second ship (at 30° angle of depression).

### Step 3: Identify the triangles
We will use two right triangles:
1. Triangle PBQ for ship B (angle of depression = 30°)
2. Triangle PAQ for ship A (angle of depression = 45°)

### Step 4: Use the tangent function to find distances
For triangle PBQ:
- The angle of depression to ship B is 30°. Therefore, angle QPB = 30°.
- The height of the cliff (PQ) is the opposite side = 200 m.
- The distance from the base of the cliff to ship B (BQ) is the adjacent side.

Using the tangent function:
\[ \tan(30°) = \frac{PQ}{BQ} \]
\[ \tan(30°) = \frac{200}{BQ} \]
We know that \(\tan(30°) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{200}{BQ} \]
Cross-multiplying gives:
\[ BQ = 200\sqrt{3} \]

### Step 5: Calculate BQ
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ BQ \approx 200 \times 1.732 \approx 346.4 \text{ m} \]

### Step 6: Calculate distance for ship A
For triangle PAQ:
- The angle of depression to ship A is 45°. Therefore, angle QPA = 45°.
- Again, the height of the cliff (PQ) is the opposite side = 200 m.
- The distance from the base of the cliff to ship A (AQ) is the adjacent side.

Using the tangent function:
\[ \tan(45°) = \frac{PQ}{AQ} \]
\[ \tan(45°) = \frac{200}{AQ} \]
We know that \(\tan(45°) = 1\):
\[ 1 = \frac{200}{AQ} \]
Cross-multiplying gives:
\[ AQ = 200 \]

### Step 7: Find the distance between the ships
The distance between the two ships (AB) is given by:
\[ AB = BQ - AQ \]
Substituting the values we found:
\[ AB = 346.4 - 200 = 146.4 \text{ m} \]

### Final Answer
The distance between the ships is **146.4 meters**.

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