Between 9:00 and 10:00, at what exact time will the hands of a clock be 180 degrees apart?

Introduction / Context:
This question is a typical clock angle problem where we must find the exact time, as a fractional number of minutes past an hour, when the angle between the hour and minute hands is 180 degrees. Such questions test your ability to translate clock positions into degrees and to use a general formula connecting the time after the hour to the angle between the clock hands. Here the time is between 9:00 and 10:00 and we are specifically looking for when the hands are in a straight line but opposite each other, forming an angle of 180 degrees.

Given Data / Assumptions:

- The time is between 9:00 and 10:00. - We want the moment when the angle between hour and minute hands is 180 degrees. - The minute hand moves 6 degrees per minute. - The hour hand moves 0.5 degrees per minute. - At exactly 9:00 the hour hand is at 270 degrees from the 12 o clock position.

Concept / Approach:
Let m be the number of minutes past 9:00. We compute the angle of the minute hand and the angle of the hour hand from the top of the clock (12 o clock position) and set their absolute difference equal to 180 degrees. The minute hand angle is given by 6 * m, and the hour hand angle at 9:m is 270 + 0.5 * m. Taking the difference and solving for m gives a linear equation. We then check which of the options matches the value of m derived from the equation.

Step-by-Step Solution:
Step 1: Let m be the minutes past 9:00. Step 2: Angle of the minute hand from 12 is 6 * m degrees. Step 3: At 9:00, the hour hand is at 9 * 30 = 270 degrees. Step 4: After m minutes, the hour hand moves an extra 0.5 * m degrees, so its angle is 270 + 0.5 * m degrees. Step 5: The angle between the hands is |270 + 0.5 * m - 6 * m| = |270 - 5.5 * m|. Step 6: Set this equal to 180 degrees: |270 - 5.5 * m| = 180. Step 7: Consider 270 - 5.5 * m = 180, which simplifies to 5.5 * m = 90 and m = 90 / 5.5 = 180 / 11 minutes. Step 8: The alternative equation 5.5 * m - 270 = 180 would give m greater than 60 minutes, which is outside the hour from 9:00 to 10:00, so it is rejected.

Verification / Alternative check:
Compute the actual angles using m = 180 over 11 minutes, which is approximately 16.36 minutes. The minute hand is then at 6 * 180 / 11 = 1080 / 11 degrees, and the hour hand is at 270 + 0.5 * 180 / 11 = 270 + 90 / 11 degrees. The difference between these two angles is exactly 180 degrees when simplified. Since m is within the 0 to 60 minute range after 9:00, this time is valid and satisfies the condition of being between 9:00 and 10:00.

Why Other Options Are Wrong:
The options involving times past 10:00 cannot be correct because the problem specifically asks for a time between 9:00 and 10:00. The other suggested fractional minutes past 9:00 do not satisfy the equation |270 - 5.5 * m| = 180 when checked. Only m equal to 180 over 11 minutes produces an exact 180 degree angle between the hands within the correct one hour interval.

Common Pitfalls:
Learners often forget that the hour hand is also moving during the minutes after 9:00 and treat it as fixed at 270 degrees, which leads to an incorrect equation. Another pitfall is failing to consider that the absolute value equation can in principle give two solutions, one of which may be outside the 0 to 60 minute range. Always set up the general formula, solve the resulting equation, and then check whether the solution falls in the specified time window.

Final Answer:
The hands of the clock will be 180 degrees apart at 180 over 11 minutes past 9:00.