Answer :
Final answer:
The minute and the hour hand of a clock form a right angle between 10 and 11 at approximately 38.18 minutes past 10.
Explanation:
The hands of a clock are at a right angle between the hours of 10 and 11 around two specific instances. To derive when this occurs, we need to be aware of how the hands move around the clock. In an hour, the minute hand makes a full rotation (360 degrees), and the hour hand moves 1/12 of that distance (or 30 degrees). In order for the two hands to form a right angle (90 degrees), the difference in their positions has to be a quarter of the clock (90 degrees). So, between 10 and 11, the minute hand has to put a distance equivalent to a quarter of the clock rotation between itself and the hour hand
Let's designate M, the angle which the minute hand covers and H the angle which the hour hand covers. Therefore we are looking for the minute - M for which, where M - H = 90 °.
The hour hand covers 30° every hour (360° / 12 hours). Since we are between 10 and 11, it has already covered 300° (30° * 10) and will add to this another 0.5° for every minute that passes (30° / 60 minutes). So H = 300° + 0.5° * m.
The minute hand covers 360° per hour, or 6° every minute (360° / 60 minutes). So, M = 6° * m.
Solving the equation M - H = 90°, we get: 6° * m - (300° + 0.5° * m) = 90°. Solving this equation, we find m is approximately 38.18 minutes, after the hour 10.
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