A clock's minute hand has length 4 and its hour hand length 3.
What is the distance between the tips at the moment when it is increasing most rapidly?
I started with theta_M(t)=pi/2 - 2pi t and theta_H(t)=pi/2 - (pi/6)t where t is in hours
d(t)=sqrt[25 - 24cos{(11/6)pi t}]
d'(t)= [22 sin{(11/6)pi t}] / d(t)
d"(t) was very tedious but setting it to zero eventually led to a quadratic expression in cos(x) where x= (11/6)pi t
(2pi - 1) (cos(x))^2 - (25 pi/12) cos(x) + 1 =0
solving this gives cos(x)= {0.1785, 1.0603} but the cosine can't be greater than 1, so only 0.1785 can be valid.
then x= 1.3913 radians; t=(6x)/(11 pi)=2.384 hours
or t is about 2:23
and d(t)=4.551
unless of course I made a math error.
And that's my story
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Posted by Larry
on 2004-03-07 02:10:02 |
my formulae for d(t), d'(t)
