What is the angular position in radians of the minute hand of a clock at 3:30, measured from the 12 o'clock position in standard position (counterclockwise from the positive $x$-axis)?

• A. $\frac{\pi }{2}$
• B. $\frac{7\pi }{6}$
• C. $\frac{5\pi }{6}$
• D. $\frac{3\pi }{2}$

Answer: (D)

Verified step by step guidance

1. Understand that the minute hand moves 360 degrees (or \$2\pi\( radians) in 60 minutes, so its angular velocity is \)\frac{2\pi}{60} = \frac{\pi}{30}$ radians per minute.
2. At 3:30, the minute hand is at the 30-minute mark, so calculate the angle by multiplying the angular velocity by the number of minutes: \$30 \times \frac{\pi}{30}$.
3. Simplify the expression to find the angular position in radians from the 12 o'clock position, which is the standard position (measured counterclockwise from the positive x-axis).
4. Recall that the angle is measured from the 12 o'clock position, so no additional adjustments are needed since the minute hand starts at 12 o'clock at 0 minutes.
5. Express the final angular position in simplest fractional form of \(\pi\), which corresponds to the position of the minute hand at 3:30.