Question 10

At $6:00$ a.m., the hour hand is at $180^\circ$ and the minute hand is at $0^\circ$.

In $m$ minutes past $6:00$:

• Hour hand moves at $0.5^\circ$ per minute: $\theta_H = 180 + 0.5m$.
• Minute hand moves at $6^\circ$ per minute: $\theta_M = 6m$.
• The angle between them is $\theta = |180 - 5.5m|$.

We set $\theta \approx 60^\circ$:

Evaluate Given Options:

• (A) 6:22 a.m. ($m = 22$):
  $$\theta = |180 - 5.5 \times 22| = |180 - 121| = 59^\circ.$$
  Difference from $60^\circ$ is $|59^\circ - 60^\circ| = 1^\circ$.
• (B) 6:27 a.m. ($m = 27$):
  $$\theta = |180 - 5.5 \times 27| = |180 - 148.5| = 31.5^\circ.$$
  Difference is $28.5^\circ$.
• (C) 6:38 a.m. ($m = 38$):
  $$\theta = |180 - 5.5 \times 38| = |180 - 209| = 29^\circ.$$
  Difference is $31^\circ$.
• (D) 6:45 a.m. ($m = 45$):
  $$\theta = |180 - 5.5 \times 45| = |180 - 247.5| = 67.5^\circ.$$
  Difference is $7.5^\circ$.

The angle is closest to $60^\circ$ at 6:22 a.m. (option A).