A man in a boat rowed away from a cliff 150 m high takes 2 min, to change the angle from `60^(@)` to `45^(@)`. The speed of the boat is

Options

A. `25(3-sqrt(3))`
B. `150(3-sqrt(3))`
C. `25(sqrt(3)-1)`
D. none of these

Answer

To find the speed of the boat, we will follow these steps:

### Step 1: Understand the Problem
We have a cliff that is 150 meters high. A man in a boat rows away from the cliff, changing his angle of elevation from 60 degrees to 45 degrees in 2 minutes. We need to find the speed of the boat.

### Step 2: Draw the Diagram
Let's label the points:
- Let point A be the top of the cliff (150 m high).
- Let point B be the base of the cliff.
- Let point C be the position of the boat when the angle is 60 degrees.
- Let point D be the position of the boat when the angle is 45 degrees.

### Step 3: Use Trigonometry to Find Distances
1. **For angle 60 degrees:**
- In triangle ABC, we can use the tangent function:
\[ \tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} \]
- Here, \( AB = 150 \) m (height of the cliff) and \( BC = x \).
- We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{150}{x} \implies x = \frac{150}{\sqrt{3}} = 50\sqrt{3} \text{ m} \]
- Therefore, \( CB = 50\sqrt{3} \text{ m} \).

2. **For angle 45 degrees:**
- In triangle ABD, we again use the tangent function:
\[ \tan(45^\circ) = \frac{AB}{BD} \]
- Here, \( BD = y \):
\[ 1 = \frac{150}{y} \implies y = 150 \text{ m} \]
- Therefore, \( DB = 150 \text{ m} \).

### Step 4: Find the Distance Traveled by the Boat
- The distance \( DC \) traveled by the boat from point C to point D is:
\[ DC = DB - CB = 150 - 50\sqrt{3} \]
- Simplifying:
\[ DC = 150 - 50\sqrt{3} \]

### Step 5: Calculate the Speed of the Boat
- The time taken is 2 minutes, which is \( 2 \times 60 = 120 \) seconds.
- The speed \( v \) is given by:
\[ v = \frac{DC}{\text{time}} = \frac{150 - 50\sqrt{3}}{120} \]
- Simplifying:
\[ v = \frac{150 - 50\sqrt{3}}{120} = \frac{25(3 - \sqrt{3})}{20} = \frac{25(3 - \sqrt{3})}{6} \]

### Final Answer
Thus, the speed of the boat is:
\[ \text{Speed} = \frac{25(3 - \sqrt{3})}{6} \text{ m/s} \]