How much time does the minute hand of a watch take to describe an angle of $\frac{3\pi}{2}$ radian ?

Question

Accepted Answer

To determine how much time the minute hand of a watch takes to describe an angle of $\frac{3\pi}{2}$ radians, we need to understand the relationship between the movement of the minute hand and the angles it describes.

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 a full circle ($2\pi$ radians) takes 60 minutes, we can set up a proportion to find out how long it takes to cover $\frac{3\pi}{2}$ radians.
   - $\frac{3\pi}{2}$ radians is $\frac{3\pi}{2\pi} = \frac{3}{4}$ of a full circle.
   - Therefore, the time taken is $\frac{3}{4} \times 60$ minutes = 45 minutes.

3. **Explanation of Why Option 2 is Correct:**
   - Option 2 states that the time taken is 45 minutes, which matches our calculation. Thus, Option 2 is the correct answer.

4. **Explanation of Why Other Options are Incorrect:**
   - **Option 1 (30 minutes):** This would imply that $\frac{3\pi}{2}$ radians is half of a full circle, which is incorrect. Half of a full circle is $\pi$ radians, not $\frac{3\pi}{2}$.
   - **Option 3 (15 minutes):** This would imply that $\frac{3\pi}{2}$ radians is a quarter of a full circle, which is incorrect. A quarter of a full circle is $\frac{\pi}{2}$ radians.
   - **Option 4 (20 minutes):** This would imply that $\frac{3\pi}{2}$ radians is one-third of a full circle, which is incorrect. One-third of a full circle is $\frac{2\pi}{3}$ radians.

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