At what time between 5:30 and 6:00 will the hands of a clock be at right angles to each other?

Introduction / Context:
This problem asks for a time between 5:30 and 6:00 when the hands of a clock are at right angles, that is, the angle between them is 90 degrees. It is more specific than the usual hour range question because it restricts the time to a half hour interval. It checks whether the candidate can correctly write the angular positions at a general time between two given points and solve for the right angle condition.

Given Data / Assumptions:

• Time interval: from 5:30 to 6:00.
• Right angle means the angle between hands is exactly 90 degrees.
• At 5:30, the hour hand is not exactly on 5 but halfway between 5 and 6.
• The minute hand moves 6 degrees per minute, hour hand 0.5 degree per minute.

Concept / Approach:
Let t be the number of minutes after 5:30. Then the time is 5:30 + t. We first compute the angle of each hand at 5:30 and then add the further movement over t minutes. We then set the absolute difference of the angles equal to 90 degrees and solve for t in the specified interval.

Step-by-Step Solution:
Step 1: Angles at 5:30. At 5:00, hour hand angle = 30 * 5 = 150 degrees. In 30 minutes, hour hand moves 0.5 * 30 = 15 degrees. So at 5:30, hour hand angle = 150 + 15 = 165 degrees. Minute hand at 5:30 is at 30 minutes, so angle = 6 * 30 = 180 degrees. Step 2: General angles after t minutes from 5:30. Hour hand angle H = 165 + 0.5 t. Minute hand angle M = 180 + 6 t. Step 3: Right angle condition. We need |M - H| = 90 degrees. Compute M - H: (180 + 6 t) - (165 + 0.5 t) = 15 + 5.5 t. Thus, |15 + 5.5 t| = 90. Since for t between 0 and 30, 15 + 5.5 t is positive, we use: 15 + 5.5 t = 90. Step 4: Solve for t. 5.5 t = 90 - 15 = 75. t = 75 / 5.5 = 150 / 11 minutes. 150 / 11 = 13 7/11 minutes. Step 5: Convert to clock time. Time = 5:30 + 13 7/11 minutes = 5:43 7/11 minutes. So the hands are at right angles at 43 7/11 minutes past 5.

Verification / Alternative check:
At t = 150 / 11: Hour hand angle = 165 + 0.5 * 150 / 11 = 165 + 75 / 11 = (1815 + 75) / 11 = 1890 / 11 degrees. Minute hand angle = 180 + 6 * 150 / 11 = 180 + 900 / 11 = (1980 + 900) / 11 = 2880 / 11 degrees. Difference = (2880 / 11) - (1890 / 11) = 990 / 11 = 90 degrees, confirming the right angle condition.

Why Other Options Are Wrong:
Option a (43 5/11 minutes) is very close but does not produce exactly 90 degrees; it is a trap for rounding errors. Options c and d are not within the specified 5:30 to 6:00 interval for right angle formation and lack any calculated basis for this time segment.

Common Pitfalls:
Some learners treat 5:30 as if the hour hand were still at 150 degrees instead of 165 degrees. Others forget to start the calculation from 5:30 and simply set time as t minutes after 5:00. Failing to restrict t to between 0 and 30 minutes can also lead to an incorrect second solution which may lie outside the allowed interval.

Final Answer:
The hands of the clock are at right angles at 43 7/11 minutes past 5.