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 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?```
 Subject: Re: calculus/derivatives/rates Answered By: secret901-ga on 02 Nov 2002 20:35 PST Rated:
 ```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: 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. 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```
 ```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```
 `I have been looking forward to the answer to this one!`