At what time between 6 𝑎.𝑚. and 7 𝑎.𝑚. will the minute hand and hour hand…

Key idea: If t is minutes after 6:00, the minute hand is at 6t degrees and the hour hand is at 180 + 0.5t degrees. The angle between them is |6t − (180 + 0.5t)| = |5.5t − 180|.

• Solve |5.5t − 180| = 60. This gives two linear equations.

• 5.5t − 180 = 60 ⇒ 5.5t = 240 ⇒ t = 240/5.5 = 480/11 ≈ 43.636 minutes ⇒ about 6:43:38.

• 5.5t − 180 = −60 ⇒ 5.5t = 120 ⇒ t = 120/5.5 = 240/11 ≈ 21.818 minutes ⇒ about 6:21:49.

Check which given time is closest to either exact solution by computing the angle at each provided time:

• At 6:22 (t = 22): angle = |5.5·22 − 180| = |121 − 180| = 59° (1° away from 60°).

• At 6:27 (t = 27): angle = |5.5·27 − 180| = |148.5 − 180| = 31.5°.

• At 6:38 (t = 38): angle = |5.5·38 − 180| = |209 − 180| = 29°.

• At 6:45 (t = 45): angle = |5.5·45 − 180| = |247.5 − 180| = 67.5° (7.5° away from 60°).

Conclusion: The times when the hands make exactly 60° between 6 and 7 are about 6:21.82 and 6:43.64. Among the given choices, 6:22 is the closest.

Answer: 6:22 a.m.