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tony123

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Registered: 2007-08-03

Posts: 229

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?

Alg Num Theory

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Registered: 2017-11-24

Posts: 693

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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

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

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Set to to 0:

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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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