The time taken by the minute hand of a watch to describe an angle of $(\frac{3\pi}{2})$ radian, is :

Question

Accepted Answer

To determine the time taken by the minute hand of a watch to describe an angle of $\frac{3\pi}{2}$ radians, we need to understand the movement of the minute hand in terms of radians and time.

1. **Understanding the Movement of the Minute Hand:**
   - The minute hand of a watch completes a full circle, which is $2\pi$ radians, in 60 minutes.

2. **Calculating the Time for $\frac{3\pi}{2}$ Radians:**
   - Since the minute hand completes $2\pi$ radians in 60 minutes, we can set up a proportion to find the time taken for $\frac{3\pi}{2}$ radians:
     $$
     \frac{2\pi \text{ radians}}{60 \text{ minutes}} = \frac{\frac{3\pi}{2} \text{ radians}}{x \text{ minutes}}
     $$
   - Solving for $x$, we cross-multiply:
     $$
     2\pi \cdot x = 60 \cdot \frac{3\pi}{2}
     $$
   - Simplifying the right side:
     $$
     2\pi \cdot x = 90\pi
     $$
   - Dividing both sides by $2\pi$:
     $$
     x = \frac{90\pi}{2\pi} = 45 \text{ minutes}
     $$

3. **Explanation of Why Option 2 is Correct:**
   - The calculation shows that the minute hand takes 45 minutes to describe an angle of $\frac{3\pi}{2}$ radians, which matches Option 2.

4. **Explanation of Why Other Options are Incorrect:**
   - **Option 1 (30 minutes):** If the minute hand took 30 minutes, it would describe an angle of $\pi$ radians, not $\frac{3\pi}{2}$ radians.
   - **Option 3 (60 minutes):** In 60 minutes, the minute hand completes a full circle of $2\pi$ radians, which is more than $\frac{3\pi}{2}$ radians.
   - **Option 4 (20 minutes):** In 20 minutes, the minute hand would describe an angle of $\frac{2\pi}{3}$ radians, which is less than $\frac{3\pi}{2}$ radians.

Therefore, the correct answer is Option 2, as the minute hand takes 45 minutes to describe an angle of $\frac{3\pi}{2}$ radians.