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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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Solution Puzzle Solution Comment 29 of 29 |
(In reply to answer by K Sengupta)

Let us respectively denote the hour hand and the minute hand by the lines CP and CQ. Since the minute hand moves faster than the hour hand, we can assume wlog that CP is fixed and CQ is moving at a constant speed. Let t denote the vector which corresponds to the velocity of the point Q under this assumption. Then, the rate of the change of the distance between P and Q is the component of t in the  direction of PQ.

Since t is orthogonal to CQ and the magnitude of t is constant, it
follows that this component is maximal whenever angle CQP = 90 degrees.

Thus, PQ = V(CP^2 - CQ^2) =  V(4^2 - 3^2) = V7 
 = 2.6457513 (correct to 7 places of decimals)

Edited on May 9, 2008, 4:11 pm
  Posted by K Sengupta on 2008-05-09 16:07:14

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