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Q: calculus/derivatives/rates ( Answered 5 out of 5 stars,   2 Comments )
Question  
Subject: calculus/derivatives/rates
Category: Reference, Education and News > Education
Asked by: jcgerry-ga
List Price: $10.00
Posted: 02 Nov 2002 18:22 PST
Expires: 02 Dec 2002 18:22 PST
Question ID: 96952
On a certain clock the minute hand is four inches long and the hour
hand is three inches long. How fast is the distance between the tips
of the hands changing at nine o'clock?
Answer  
Subject: Re: calculus/derivatives/rates
Answered By: secret901-ga on 02 Nov 2002 20:35 PST
Rated:5 out of 5 stars
 
Hello jcgerry,
Thank you for the interesting question.  I shall explain how I solved
it.

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
Pythagorean theorem.

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:
r = 3 + 4 - 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.
secret901-ga

Search strategy:
Law of cosines

Clarification of Answer by secret901-ga on 02 Nov 2002 20:39 PST
Please forgive me.  The rate of revolution of the hour hand is
actually 2pi/12 hours = pi/6 /hour.  The rest of the computation is
correct.  I just mistyped it.
secret901-ga
jcgerry-ga rated this answer:5 out of 5 stars

Comments  
Subject: Re: calculus/derivatives/rates
From: secret901-ga on 02 Nov 2002 20:39 PST
 
Please forgive me.  The rate of revolution of the hour hand is
actually 2pi/12 hours = pi/6 /hour.  The rest of the computation is
correct.  I just mistyped it.
secret901-ga
Subject: Re: calculus/derivatives/rates
From: gan-ga on 03 Nov 2002 01:05 PST
 
I have been looking forward to the answer to this one!

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