Between 3 o clock and 4 o clock, at what exact time will the hour hand and the minute hand be on the same straight line but facing in opposite directions (forming a 180 degree angle)?

Introduction / Context:
This problem is about the hour and minute hands being in a straight line but pointing in opposite directions, so that the angle between them is 180 degrees. Unlike coincidence problems where the angle is 0, here we want the hands to form a straight line between 3 o clock and 4 o clock. This involves setting the angular difference between the hands equal to 180 degrees and solving for the time.

Given Data / Assumptions:

• We are considering the interval between 3 o clock and 4 o clock.
• At 3 o clock, the hour hand is on the 3 and the minute hand is on the 12.
• The minute hand moves 6 degrees per minute.
• The hour hand moves 0.5 degrees per minute.
• We require the angle between the two hands to be 180 degrees.

Concept / Approach:
At any time t minutes after 3 o clock, the angle of the minute hand and the angle of the hour hand can be expressed from the 12 o clock position. The angle between the hands is the absolute difference of these two angles. For the hands to be in a straight line but opposite each other, this difference must be 180 degrees. We set up an equation based on this condition and solve for t to find the required time.

Step-by-Step Solution:
Step 1: At 3 o clock, the hour hand angle from 12 is 3 * 30 = 90 degrees. Step 2: The minute hand angle from 12 at exactly 3 o clock is 0 degrees. Step 3: After t minutes, the minute hand angle is 6t degrees. Step 4: After t minutes, the hour hand angle is 90 + 0.5t degrees. Step 5: The angle between the hands is |6t - (90 + 0.5t)| degrees. Step 6: For the hands to be opposite each other, set this equal to 180 degrees: 6t - (90 + 0.5t) = 180. Step 7: Simplify: 6t - 90 - 0.5t = 180, so 5.5t - 90 = 180. Step 8: Solve for t: 5.5t = 270, so t = 270 / 5.5 = 540 / 11 minutes. Step 9: Convert 540 / 11 to a mixed number. 11 goes into 540 forty nine times (49 * 11 = 539) with a remainder of 1, so t = 49 1/11 minutes. Step 10: The required time is therefore 3:49 1/11.

Verification / Alternative check:
We can check quickly by computing the angles at t = 49 1/11 minutes. The minute hand angle is 6 * (540 / 11) = 3240 / 11 degrees. The hour hand angle is 90 + 0.5 * (540 / 11) = 90 + 270 / 11 = (990 + 270) / 11 = 1260 / 11 degrees. The difference is (3240 - 1260) / 11 = 1980 / 11 = 180 degrees exactly. This confirms that at 3:49 1/11 the hands are 180 degrees apart and therefore lie on the same straight line in opposite directions.

Why Other Options Are Wrong:
3:15 2/8: This is much earlier than the correct time and the hands are far from being opposite.

3:49: This is close but slightly earlier; the angle between the hands will be slightly less than 180 degrees.

3:51: Later than the correct time, by which point the minute hand angle has gone past the exact 180 degree separation.

3:45: A commonly guessed time, but at 3:45 the angle between the hands is not exactly 180 degrees.

Common Pitfalls:
Common mistakes include using 90 degrees instead of 180 degrees in the equation, which corresponds to a right angle rather than a straight line, or neglecting to take the absolute value of the angle difference. Some students also stop as soon as they find a value that looks close, such as 3:49, without converting the exact fraction or verifying the result by substituting it back into the angle expressions.

Final Answer:
The hands are on the same straight line but opposite each other at 3:49 1/11.