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in how many minutes after four o'clock will the minute hand coincides with the hour hand ?

Measure angles with respect to the positive y-axis, increasing in the clockwise direction.

At four o'clock, the angle of the minute hand is 0.

The minute hand travels 2π radians per hour = π/30 radian per minute.

At t minutes after four o'clock, the angle of the minute hand is tπ/30.

At four o'clock, the angle of the hour hand is 2π/3.

The hour hand travels 2π/12 radian per hour = π/360 radian per minute.

At t minutes after four o'clock, the angle of the hour hand is (2π/3) + (tπ/360).

Solve for t:

tπ/30 = (2π/3) + (tπ/360)

12tπ/360 = 240π/360 + tπ/360

12tπ = 240π + tπ

12t = 240 + t

11t = 240

t = 240/11≅ 21.8 minutes

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