let D = distance between the tips in cm
let a = angle between the hands
using the law of cosines ...
D^2 = 11^2 + 5^2 - 2(11)(5)cos(a)
take the derivative of the above equation w/r to time, t ...
2D(dD/dt) = 110sin(a)*(da/dt)
dD/dt = 110sin(a)*(da/dt)/(2D)
now, at 2 o'clock ...
a = 1/6 of the clock circle = pi/3 radians
rate of change of the minute hand's angle = 2pi radians in 60 minutes =
(pi/30) rad/min
rate of change of the hour hand's angle = 2pi radians in 12 hours =
(pi/6)rad/hr = (pi/360) rad/min
so, at 2 o'clock, da/dt = (pi/30) rad/min - (pi/360) rad/min = (11pi/360) rad/min
also, at 2 o'clock, D = sqrt[146 - 110cos(pi/3)]
punching all this stuff into a calculator yields ...
dD/dt = approx 0.4793 cm/min
