Opposite directions between 2 and 3 o’clock: At what time between 2:00 and 3:00 will the clock hands be in opposite directions (180° apart)?

Introduction / Context:We again use relative angular speed to find when the hands are 180° apart, now between 2 and 3.

Given Data / Assumptions:

• At 2:00, hour hand = 60°.
• Hour hand speed = 0.5°/min; minute hand speed = 6°/min.

Concept / Approach:Angle difference after t minutes: |(60 + 0.5t) − 6t| = |60 − 5.5t|. Set equal to 180 and solve in 0 < t < 60.

Step-by-Step Solution:1) |60 − 5.5t| = 180.2) 60 − 5.5t = −180 ⇒ 5.5t = 240 ⇒ t = 240/5.5 = 480/11 minutes.3) The other case gives negative t; discard.

Verification / Alternative check:t ≈ 43 7/11 min; substituting yields a 180° separation.

Why Other Options Are Wrong:Other fractions do not satisfy |60 − 5.5t| = 180 within the hour.

Common Pitfalls:Mixing 90° vs 180°; ignoring the hour-hand movement.

Final Answer:480/11 min past 2