Question 395514
We can assign 12:00 as 0 degrees, 1:00 as 30 degrees, etc.


If the time is 5 hours and x minutes after 5:00, then the angle formed by the minute hand is (x/60)(360) = 6x (degrees). For each minute after 5, the hour hand moves 1/2 degree, so the hour hand has an angle of 150 + x/2. The angle between them is 90 degrees, so


{{{abs((150 + x/2) - 6x) = 90}}}


{{{abs(150 - 11x/2) = 90}}}


{{{150 - 11x/2 = 90}}} or {{{150 - 11x/2 = -90}}}


{{{x = 120/11 = 10.909}}} or {{{x = 480/11 = 43.636}}}


Therefore the angle between the hands is 90 at roughly 5:11 and 5:44.