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Find the maximum height (in exact value) of a cylinder of radius $x$ so that it can completely place into a $100$ cm $\times$ $60$ cm $\times$ $50$ cm cubiod.

This question comes from http://hk.knowledge.yahoo.com/question/question?qid=7012072800395.

I know that this question is equivalent to two times of the maximum height (in exact value) of a right cone of radius $x$ so that it can completely place into a $50$ cm $\times$ $30$ cm $\times$ $25$ cm cubiod whose the apex of the right cone is placed at the corner of the cubiod, but I still have no idea until now.

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Please don't put subjective assessments of the difficulty of the question, such as "tricky" or "challenging", in the title. – joriki Aug 4 '12 at 20:59
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Why not? It is no doubt that this question is challenging for most people. Moreover, does the title really break the rule in MSE? – doraemonpaul Aug 4 '12 at 21:09
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No, no rule, I just kindly asked you not to do it :-) But seriously, if you've given it serious consideration and you still think there's reason to believe that most people will find this challenging, and you feel that that's valuable information to have in the title, I guess that's OK; but I keep seeing questions entitled "tricky X" and in most cases it's obvious that the OP has no idea what's tricky for other people. – joriki Aug 4 '12 at 21:36
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What have you tried? – Code-Guru Aug 4 '12 at 21:58
looks trivial to me! – Mercy Aug 5 '12 at 15:43
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Does the $x$ axis-aligned? Would it possible to have $x=0$? The only thing want to mention is the maximum length of a line within the bounded space would be the diagonal, based on the metric property.

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If you mean that the cylinder's flat faces have to be parallel to a side of the box, the question would be trivial, since for each of the three cases the max r is when the cylinder extends all the way between the faces of the box. Your statement about the max r is correct: the length of the diagonal is an (unachievable) max r for actual nondegenerate cylinders, and to approach that max r would mean using small x going to 0. Still, seems more work to do to get max r as a function of x, likely a piecewise function. – coffeemath Oct 18 '12 at 18:47

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