Thank you for the interesting question. I shall explain how I solved
First off, let us see what the clock looks like at 9 o'clock. At that
point in time, the minute hand will be at 12 and the hour hand will be
at 9, making the angle between them to be pi/2. In addition, the
distance between the tips of the hands will be 5, according to the
Let's call the angle between the two hands at any given point theta,
and the distance between their tips at any given point r.
Since the minute hand makes a complete revolution around the clock
every one hour, its rate of rotation is 2pi/hr. Similarly, the hour
hand makes a complete revolution around the clock every 12 hours,
making its rate of revolution to be 2pi/6 hr = pi/3 / hr.
Since the minute and hour hands move in the same direction (namely,
clockwise), the rate of change for the angle between them is the
difference of their rates of revolution: dtheta/dt = 2pi/hr - pi/3 /hr
= 11pi/6 / hr.
Now, according to the law of cosines
(http://hyperphysics.phy-astr.gsu.edu/hbase/lcos.html), the distance
between the tips of the two hands is given by this formula:
r² = 3² + 4² - 2(3)(4) cos (theta) = 25 - 24 cos (theta)
Now, differentiating both sides with respect to t, we get:
2r dr/dt = 24 sin (theta) dtheta/dt
dr/dt = [24 sin(theta) dtheta/dt]/2r
Now, pluggling in the values at 9 o'clock, we get:
dr/dt = [24 sin(pi/2) 11pi/6]/(2*5)
dr/dt = 44 pi/10 in/hr
dr/dt = 4.4 pi in/hr
So, the answer to your question is that at 9 o'clock, the rate of
change of the tips of the hands is about 13.8 in/hr or 0.23 in/minute.
I hope that answers your question. If there is something you don't
understand, please request for clarification before rating this answer
and I'll be happy to explain it.
Law of cosines