Question

Kumbhakarna starts sleeping between 1 pm and 2 pm and he wakes up when his watch shows such a time that the two hands (i.e., hour-hand and minute-hand) interchange the respective places. He wakes up between 2 pm and 3 PM on the same night. How long does he sleep ?​

Answer 1

Answer:

55 Mins

Step-by-step explanation:

Kumbhakarna starts sleeping between 1 pm and 2 pm and he wakes up when his watch shows such a time that the two hands (i.e., hour-hand and minute-hand) interchange the respective places. He wakes up between 2 pm and 3 PM on the same night.

Kumbhakarna starts sleeping between 1 pm and 2 pm

so Hour hand between 1 & 2

He wakes up between 2 pm and 3 PM

so now Hour Hand is Between 2 & 3  & minute Hand is between 1 & 2

( as Minute & hour hand interchanged their position)

Hence Minute hand should be between 2  & 3 Between ( 1 - 2 pm)

for every 360° rotation(60 minutes) of  minute hand hour hand moves 30°

=> for every 12° rotation (1 minutes) of  minute hand hour hand moves 1°

Minute hand between 2 & 3 Means ( 60° to 90°)

Hence hour hand =  ( 5° to 7.5°) after 1 PM (30 °)

Let say minute hand is at 60° + x°  ( x < 30°)

then hour hand = 5° + (x/12)°  after 1 PM (30 °)

= 35° + (x/12)°

This mean minute hand will move

360° - (60° + x°) + 35° + (x/12)°  to reach earlier Hour hand

= 335° + 11x/12°

This movement means Hour hand will move

= (335° + 11x/12°)/12

Hour hand will move from 35° + (x/12)°   to 60° + x°

= (25 + 11x/12)°

Equating hour hand movement

(335° + 11x/12°)/12 = (25 + 11x/12)°

=> 335° + 11x/12° = 300 + 11x

=> 35° =  121x/12

=> x = 3.47°

3.47° = 0.58 minutes

Initial position was 1 PM 10.58 minutes

Final position was  2PM 5.88 minutes

He slept around 55.3 minutes

If We take it simple

Then he slept at 1 : 10 PM

& waked up at 2: 05 PM

Slept for 55 Mins

Answer 2

Answer:

Step-by-step explanation:

I assume you know the formula for angle between 2 hands of a clock

i.e. mod(30*1 - (11/2)*12)

Now lets assume a time between 1PM and 2Pm, 1:12PM

the angle between the hands will be

mod(30*1 - (11/2) *12)

=36 degree

Now the hands are interchanged, but the angle will remain constant

again applying the formula

hour hand = 2

minute hand = ?

theta = 36

36 = (30*2 - 11/2 * m)

24 = 11/2 m

m = 4.3

now the time he slept is from 1:12 - 2:4.3

i.e  for 52.3 minutes