Find the times between 4 o'clock and 5o'clock when the angle between the minute-hand and hour-hand is `(8pi)/(15)` radians.

Explanation

`(8pi)/(15) "radians" =(8xx180^(@))/(15)=96^(@)`
We know that the circumference of clock is equal to 60 minute divisions.
`therefore 360^(@)` angle is subtended by 60 minute devisions
`therefore 1^(@) " " " " (60)/(360^(@)) " "`
`therefore 96^(@) " " " " (60xx96^(@))/(360^(@)) " "`
=16 minute divisions
`therefore` The distance between two hands will be 6 minute divisions. Now, at 4 o'clock the minute hand was 20 minute divisions behind the hour-hand. So, the minute-hand gains either (20-16)=4 minuter divisions or (20+16)=36 minute divisions. We know that the minute-hand gains 55 mintute divisions in 60 minutes
`therefore` the minute-hand gains 4 minun te divisions in ` (60xx4)/(55) "minutes"=(48)/(11) "minutes "=4(4)/(11)` minutes
Also, the minute hand gains 36 mnute divisions in `(60xx36)/(55) "minutes"=39(3)/(11)` minutes
Hence the required time will be `4(4)/(11) "minutes past 4 o'clock and "39(3)/(11)` minutes past 4 o'clock.