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 how much more time it will take to reach the shore.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
A man on a cliff observes a boat at an angle of depression of \(30^\circ\) and then three minutes later, the angle of depression is \(60^\circ\). We need to determine how much more time it will take for the boat to reach the shore.

### Step 2: Draw the Diagram
Draw a diagram to visualize the situation:
- Let the height of the cliff be \(h\).
- Let the initial position of the boat be point \(B\) and the position after 3 minutes be point \(C\).
- The point directly beneath the man on the cliff is point \(A\) (the shore).
- The angles of depression from the man to the boat are \(30^\circ\) and \(60^\circ\).

### Step 3: Set Up the Triangles
Using trigonometry:
1. For the first position of the boat (angle \(30^\circ\)):
\[ \tan(30^\circ) = \frac{h}{D_1} \quad \Rightarrow \quad D_1 = \frac{h}{\tan(30^\circ)} = h \sqrt{3} \]

2. For the second position of the boat (angle \(60^\circ\)):
\[ \tan(60^\circ) = \frac{h}{D_2} \quad \Rightarrow \quad D_2 = \frac{h}{\tan(60^\circ)} = \frac{h}{\sqrt{3}} \]

### Step 4: Relate Distances and Time
The distance the boat travels in 3 minutes is:
\[ D_1 - D_2 = 3U \]
Where \(U\) is the speed of the boat.

Substituting the expressions for \(D_1\) and \(D_2\):
\[ h \sqrt{3} - \frac{h}{\sqrt{3}} = 3U \]
Multiply through by \(\sqrt{3}\) to eliminate the fraction:
\[ 3h - h = 3U\sqrt{3} \]
This simplifies to:
\[ 2h = 3U\sqrt{3} \quad \Rightarrow \quad h = \frac{3U\sqrt{3}}{2} \]

### Step 5: Find the Remaining Distance
Now, we need to find how much more time it will take for the boat to reach the shore from point \(C\):
\[ D_2 = \frac{h}{\sqrt{3}} = \frac{3U\sqrt{3}}{2\sqrt{3}} = \frac{3U}{2} \]

### Step 6: Calculate Time to Reach Shore
The time \(T\) taken to travel distance \(D_2\) at speed \(U\) is:
\[ T = \frac{D_2}{U} = \frac{\frac{3U}{2}}{U} = \frac{3}{2} \text{ minutes} \]

### Step 7: Conclusion
Thus, the boat will take an additional \(1.5\) minutes to reach the shore.

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