Law of Cosine
#c^2=a^2+b^2-2ab cos theta#
Let #c# be the distance between the tips, and let #theta# be the angle between those hands. Also, let #a=4# and #b=9#.
So, we have the equation
#c^2=4^2+9^2-2cdot4cdot9 cos theta=97-72cos theta#
At 2:10pm, #theta=pi/3#
#Rightarrow c^2=97-72cos(pi/3)=61 Rightarrow c=sqrt{61}#
Since the second hand turns at #-2pi# rad/min, and the minute hand turns at #-{pi}/30# rad/min,
#{d theta}/{dt}=-2pi-(-pi/30)=-{59pi}/{30}# rad/min
By Implicit Differentiation with respect to #t#,
#c^2=97-72cos theta Rightarrow 2c{dc}/{dt}=72sin theta cdot {d theta}/{dt}#
by dividing by #2c#,
#Rightarrow {dc}/{dt}={36sin theta}/c cdot{d theta}/{dt}={36sin(pi/3)}/{sqrt{61}}(-{59pi}/30)#
#=-{177pi}/5sqrt{3/{61}}# mm/min
I hope that this was helpful.
