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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 Spreadsheet solution. | Comment 1 of 29

čIf d is the distance between the tips, then

d² = 4² + 3² - 2*4*3*cosč  by the law of cosines.

That comes out to (25 - 24 cosč)^(½)

Its derivative is ½ (25 - 24 cosč)^(-½)*24*sinč

The second derivative has in the numerator (apart from a factor of 12):

cosč(25 - 24 cosč)^(½) - 12*sin²č/(25 - 24 cosč)^(½)

In an Excel spreadsheet, if č is to be in cell B1, then d can be placed in B2 with formula =(25-24*COS(B1))^0.5, and its derivative in B3 with =12*SIN(B1)/B2. To make the second derivative zero, we put the two parts of its numerator in B4 and C4 (=COS(B1)*B2 and =12*(SIN(B1))^2/B2).  In D4 we put B4-C4, and used the solver to equate the latter to zero.

The derivative of d with respect to č is at its maximum, 3 units per radian, when the distance, d, is <FONT face=Arial size=2>2.645751311 and the separation is <FONT face=Arial size=2>0.722734248 radians, or <FONT face=Arial size=2>41.40962211 degrees.

  Posted by Charlie on 2004-02-27 14:43:07
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