At what time will the hands of a clock be at right angles after 2:00?

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Question: At what time will the hands of a clock be at right angles after 2:00?

Options:

Correct Answer: 2:30

Solution:

The hands of a clock are at right angles at 2:30.

At what time will the hands of a clock be at right angles after 2:00?

At what time will the hands of a clock be at right angles after 2:00?

Step 1: Understand that a clock has two hands: the hour hand and the minute hand.
Step 2: Know that the hour hand moves slowly and the minute hand moves quickly.
Step 3: At 2:00, the hour hand is pointing at 2 and the minute hand is pointing at 12.
Step 4: Right angles are 90 degrees apart. We need to find when the hands are 90 degrees apart.
Step 5: The minute hand moves 6 degrees for every minute (360 degrees / 60 minutes).
Step 6: The hour hand moves 0.5 degrees for every minute (30 degrees / 60 minutes).
Step 7: At 2:00, the hour hand is at 60 degrees (2 x 30 degrees).
Step 8: Let 'm' be the number of minutes after 2:00.
Step 9: The position of the minute hand after 'm' minutes is 6m degrees.
Step 10: The position of the hour hand after 'm' minutes is 60 + 0.5m degrees.
Step 11: Set up the equation for right angles: |(60 + 0.5m) - (6m)| = 90.
Step 12: Solve the equation: 60 + 0.5m - 6m = 90 or 6m - (60 + 0.5m) = 90.
Step 13: Simplify the first equation: -5.5m = 30, so m = 30/5.5 = 5.45 minutes.
Step 14: Simplify the second equation: 5.5m = 150, so m = 150/5.5 = 27.27 minutes.
Step 15: The two solutions are approximately 5.45 minutes and 27.27 minutes after 2:00.
Step 16: The first solution (5.45 minutes) is not valid since it is before 2:30.
Step 17: The valid solution is 27.27 minutes, which is approximately 30 minutes after 2:00.
Step 18: Therefore, the hands of the clock will be at right angles at 2:30.

Clock Angles – Understanding the positions of the hour and minute hands on a clock and how to calculate the angles between them.
Time Calculation – Calculating specific times based on the movement of clock hands and their relative positions.