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Question

Difficulty: EasyClocks and Calendars

A standard analog clock indicates the time is exactly 4:10. Calculate the smaller angle, in degrees, formed between the hour hand and the minute hand.

Answer: 65 / 65° / 65 degrees

Answer

65 degrees
At exactly 4:10, the minute hand has moved 60∘60^\circ from the top of the clock, and the hour hand has moved 120∘120^\circ for the four hours plus an extra 5∘5^\circ for the ten minutes passed. The difference between their specific positions (125∘125^\circ and 60∘60^\circ) is exactly 65∘65^\circ.

Step-by-Step Solution

1
Calculate the angular position of the minute hand relative to the 12 o'clock mark.
The minute hand is at 10 minutes, which corresponds to an angle of 10×6∘=60∘10 \times 6^\circ = 60^\circ.
The minute hand moves 360∘360^\circ in 60 minutes, which equals a speed of 6∘6^\circ per minute.
2
Calculate the angular position of the hour hand relative to the 12 o'clock mark.
The hour hand has moved past the 4 o'clock mark. Its position is 4×30∘+10×0.5∘=120∘+5∘=125∘4 \times 30^\circ + 10 \times 0.5^\circ = 120^\circ + 5^\circ = 125^\circ.
The hour hand moves 30∘30^\circ per hour and an additional 0.5∘0.5^\circ per minute due to continuous drift.
3
Find the absolute difference between the two angular positions to determine the angle between the hands.
∣125∘−60∘∣=65∘|125^\circ - 60^\circ| = 65^\circ.
The angle between the two hands is simply the absolute difference of their individual positions from the 12 o'clock reference point.

Key Concept

Clock hand angular positioning and the calculation of hour-hand drift.
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Topics

Clocks and Calendars

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