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#1 2018-06-02 00:03:52

tony123
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Registered: 2007-08-03
Posts: 196

highest position

An insect sets off upwards from the shaft of the minute hand of a church-clock exactly at 12 o'clock. Moving uniformly along the hand, it reaches the end of the hand in a quarter of an hour. When was it at the highest position?

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#2 2018-06-02 23:20:53

Alg Num Theory
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Registered: 2017-11-24
Posts: 339
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Re: highest position

Let the insect’s speed along the minute hand be v. After time t, the insect has moved a distance vt from the base of the hand and so its height h above the horizontal through the centre of the clock face is given by

(note that the angular velocity of the minute hand is 2π/(60×60) radians per second). Differentiate:

Set to to 0:

This equation can’t be solved analytically; using Wolfram|Alpha, I get t ≈ 493 seconds = 8 minutes 13 seconds, i.e. the insect is at its highest position at 12:08:13.

Check that d²h/dt² < 0 so that this is indeed a maximum value.

Last edited by Alg Num Theory (2018-06-02 23:24:10)

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