A man on a cliff observes a boat, at an angle of depression `30^(@)` , which is sailing towards the shore to the point immediately beneath him. Three minutes later, the angle of depression of the boat is found to be `60^(@)` . Assuming that the boat sails at a uniform speed, determine.
the speed of the boat in metre per second if the height of the cliff is 500 m

Answer

To solve the problem step by step, we will use trigonometric concepts and the relationships between angles, distances, and heights.

### Step 1: Understand the Problem
We have a cliff of height \( H = 500 \) m. A man on the cliff observes a boat at two different angles of depression: first at \( 30^\circ \) and then at \( 60^\circ \) after 3 minutes. We need to find the speed of the boat in meters per second.

### Step 2: Draw the Diagram
Draw a right triangle where:
- Point A is the top of the cliff.
- Point B is the point directly below A on the water (the shore).
- Point C is the position of the boat when the angle of depression is \( 30^\circ \).
- Point D is the position of the boat when the angle of depression is \( 60^\circ \).

### Step 3: Set Up the Relationships
Using the angles of depression:
- For angle \( 30^\circ \):
\[ \tan(30^\circ) = \frac{H}{BC} \implies \frac{1}{\sqrt{3}} = \frac{500}{BC} \implies BC = 500\sqrt{3} \]

- For angle \( 60^\circ \):
\[ \tan(60^\circ) = \frac{H}{BD} \implies \sqrt{3} = \frac{500}{BD} \implies BD = \frac{500}{\sqrt{3}} \]

### Step 4: Calculate Distances
Let \( BC \) be the distance from the boat's position when the angle is \( 30^\circ \) to the point directly beneath the cliff, and \( BD \) be the distance when the angle is \( 60^\circ \).

From the above calculations:
- \( BC = 500\sqrt{3} \)
- \( BD = \frac{500}{\sqrt{3}} \)

### Step 5: Find the Distance the Boat Sailed
The distance \( CD \) that the boat has sailed in 3 minutes is:
\[ CD = BC - BD = 500\sqrt{3} - \frac{500}{\sqrt{3}} \]

### Step 6: Simplify the Expression
To simplify \( CD \):
\[ CD = 500\left(\sqrt{3} - \frac{1}{\sqrt{3}}\right) = 500\left(\frac{3 - 1}{\sqrt{3}}\right) = 500\left(\frac{2}{\sqrt{3}}\right) = \frac{1000}{\sqrt{3}} \text{ m} \]

### Step 7: Calculate the Speed of the Boat
The speed of the boat \( v \) is given by:
\[ v = \frac{CD}{\text{time}} = \frac{\frac{1000}{\sqrt{3}}}{3 \text{ minutes}} = \frac{1000}{3\sqrt{3}} \text{ m/min} \]

### Step 8: Convert Speed to m/s
To convert from meters per minute to meters per second:
\[ v = \frac{1000}{3\sqrt{3}} \times \frac{1}{60} \text{ m/s} \]
Calculating this gives:
\[ v \approx \frac{1000}{3 \times 1.732 \times 60} \approx \frac{1000}{311.04} \approx 3.21 \text{ m/s} \]

### Final Answer
Thus, the speed of the boat is approximately \( 3.21 \) m/s.