Question

clock problem:
how soon after 1 o clock will the hands of a clock form a right angle

Answer 1

the minute hand rotates once per hour, so it moves 360/60 = 6°min
The hour hand takes 12 hours to go around, so it moves 360/720 = 1/2 °/min
So you want t minutes, such that
0 + 6t = 30 + 1/2 t + 90
Now solve for t

Answer 2

Consider noon to correspond with 0° (standard "bearing")
and consider the 1:00 position.
The angle formed by the hour hand between noon and 1:00 is 30°
and the angle formed by the minute hand is 0°
after a time covered by x°, the hour hand is at 30+x/60 °
and the minute hand is at x °

But we want
x - (30+x/60) = 90
59x/60 = 120
x = 122.0339 °
so the minute hand is at the 122.0339° position

122.0339/360 = minute/60
minutes = 20.339 or 20 minutes, 20 seconds

which puts the time at 1:20:20

check my arithmetic