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**tony123****Member**- Registered: 2007-08-03
- Posts: 197

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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**Alg Num Theory****Member**- Registered: 2017-11-24
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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 2*π*/(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 d²*h*/d*t*² < 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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