From the top of a light house 100 m high, t he angles of depression of two ships are observed as `48^(@) and 36^(@)` respectively. Find the distance between the two ships (in the nearest metre ) if :
the ships are on the same side of the light house.

Answer

To solve the problem of finding the distance between two ships observed from the top of a lighthouse, we can follow these steps:

### Step 1: Understand the Problem
We have a lighthouse that is 100 m high. The angles of depression to two ships are given as 48° and 36°. We need to find the distance between the two ships, assuming they are on the same side of the lighthouse.

### Step 2: Draw the Diagram
1. Draw a vertical line representing the lighthouse, labeled as AB, where AB = 100 m (height of the lighthouse).
2. Mark the position of the two ships, C and D, on the same side of the lighthouse.
3. Draw horizontal lines from points C and D to the top of the lighthouse (point A).
4. Mark the angles of depression: ∠ACB = 48° and ∠ADB = 36°.

### Step 3: Identify the Triangles
We will analyze two right triangles:
- Triangle ABC (for angle 48°)
- Triangle ABD (for angle 36°)

### Step 4: Calculate the Distances
1. **For Triangle ABC (angle 48°)**:
- Using the tangent function:
\[ \tan(48°) = \frac{AB}{BC} \]
- Here, AB = 100 m (height of the lighthouse), and BC is the distance from the lighthouse to ship C.
- Rearranging gives:
\[ BC = \frac{AB}{\tan(48°)} = \frac{100}{\tan(48°)} \]
- Using a calculator, we find:
\[ \tan(48°) \approx 1.1106 \]
- Thus,
\[ BC \approx \frac{100}{1.1106} \approx 90.04 \text{ m} \]

2. **For Triangle ABD (angle 36°)**:
- Using the tangent function again:
\[ \tan(36°) = \frac{AB}{BD} \]
- Rearranging gives:
\[ BD = \frac{AB}{\tan(36°)} = \frac{100}{\tan(36°)} \]
- Using a calculator, we find:
\[ \tan(36°) \approx 0.7265 \]
- Thus,
\[ BD \approx \frac{100}{0.7265} \approx 137.64 \text{ m} \]

### Step 5: Find the Distance Between the Ships
Since both ships are on the same side of the lighthouse, the distance between the two ships (CD) is given by:
\[ CD = BD - BC \]
Substituting the values we found:
\[ CD \approx 137.64 \text{ m} - 90.04 \text{ m} \approx 47.6 \text{ m} \]

### Final Answer
The distance between the two ships is approximately **48 m** (rounded to the nearest metre).

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