An observer on the top sea 500 meter high level , observes the angles of depression of the two boats in his same place of vision to be `45^(@)and30^(@)` respectively .Then the distance between the boats , if the boats are on the same side of the mountain , is

Options

A. 456 m
B. 584 m
C. 366 m
D. 699 m

Answer

To solve the problem step by step, we will use trigonometric ratios and the properties of right-angled triangles.

### Step 1: Understand the Problem
We have an observer at the top of a 500-meter high mountain. The angles of depression to two boats are given as 45° and 30°. We need to find the distance between the two boats.

### Step 2: Draw the Diagram
1. Draw a vertical line representing the mountain, with point A at the top (500 m high).
2. Draw a horizontal line from point A to represent the line of sight to the boats.
3. Mark the positions of the two boats as points B and C, where the angle of depression to boat B is 45° and to boat C is 30°.

### Step 3: Set Up the Right Triangles
- For boat B (angle of depression = 45°):
- Let D be the point directly below A on the ground.
- Triangle ABD is a right triangle where:
- AD = 500 m (height of the mountain)
- Angle ADB = 45°

- For boat C (angle of depression = 30°):
- Triangle ACD is another right triangle where:
- AC = 500 m (height of the mountain)
- Angle ADC = 30°

### Step 4: Calculate Distances Using Trigonometry
1. **For Boat B**:
- Using the tangent function:
\[ \tan(45°) = \frac{AD}{DB} \]
\[ 1 = \frac{500}{DB} \implies DB = 500 \text{ m} \]

2. **For Boat C**:
- Again using the tangent function:
\[ \tan(30°) = \frac{AD}{DC} \]
\[ \frac{1}{\sqrt{3}} = \frac{500}{DC} \implies DC = 500\sqrt{3} \text{ m} \]

### Step 5: Find the Distance Between the Boats
- The total distance between the two boats (BC) can be calculated as:
\[ BC = DC - DB \]
\[ BC = 500\sqrt{3} - 500 \]
\[ BC = 500(\sqrt{3} - 1) \]

### Step 6: Substitute the Value of \(\sqrt{3}\)
- Using \(\sqrt{3} \approx 1.732\):
\[ BC = 500(1.732 - 1) = 500(0.732) \approx 366 \text{ m} \]

### Final Answer
The distance between the two boats is approximately **366 meters**.

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