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Question

Difficulty: MediumClocks and Calendars

At exactly 4:284:28 PM, what is the exact acute angle (in degrees) between the hour hand and the minute hand of a standard analog clock?

Answer: 34 degrees

Answer

34
The angle is determined by finding the exact position of both hands relative to the 12 o'clock mark. The minute hand is at 168 degrees (28 minutes multiplied by 6 degrees). The hour hand is at 134 degrees, which includes its base position at 4 o'clock (120 degrees) plus the drift from the 28 minutes that have passed (14 degrees). The absolute difference between these two positions is exactly 34 degrees.

Step-by-Step Solution

1
Calculate the angle of the minute hand from the 12 o'clock position.
28×6∘=168∘28 \times 6^\circ = 168^\circ
The minute hand moves at a rate of 6 degrees per minute.
2
Calculate the base angle of the hour hand exactly at 4 o'clock.
4×30∘=120∘4 \times 30^\circ = 120^\circ
Each hour marking on the clock represents 30 degrees from the top.
3
Calculate the drift of the hour hand caused by the 28 elapsed minutes.
28×0.5∘=14∘28 \times 0.5^\circ = 14^\circ
The hour hand continuously moves at a rate of 0.5 degrees per minute.
4
Determine the true position of the hour hand.
120∘+14∘=134∘120^\circ + 14^\circ = 134^\circ
Adding the drift to the base position gives the exact current location of the hour hand.
5
Calculate the acute angle between the two hands.
∣168∘−134∘∣=34∘|168^\circ - 134^\circ| = 34^\circ
The angle between the hands is the absolute difference between their positions from the 12 o'clock mark.

Key Concept

Calculating the precise angle between clock hands by accounting for hour-hand drift
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Topics

Clocks and Calendars

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