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Like Clockwork (Posted on 2004-02-27) Difficulty: 4 of 5
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?

See The Solution Submitted by DJ    
Rating: 4.0000 (9 votes)

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re(2): why calculus | Comment 21 of 29 |
(In reply to re: why calculus by Steve)

I must agree with Axorion/Dan here.  The fact that the hour hand moves is irrelevant because we only need to consider the relative movement of the hands.  We can say that relative to the minute hand, the hour hand is moving counter-clockwise around a point 4 inches away from the point of the minute hand (the center of the clock).

Relative to the minute hand, the hour hand keeps the same speed.  The time when the distance is increasing most is when the hour hand is moving in the opposite direction.  The direction is determined by the line tangent to the circle, at the point of the hour hand.  So at this moment, the point of the minute hand is on the tangent line.  So, using the pythagorean theorum, the distance is (4^2-3^2) or 7.

It seems to me like what you're proposing is just a less straightforward way (but I could be wrong).  I hope I was thorough enough and not too thorough. 

  Posted by Tristan on 2004-03-03 20:05:41
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