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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 $100 cm \times 60 cm \times50 cm$ cuboid.

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 $50 cm \times$ $30 cm \times 25 cm$ cuboid whose the apex of the right cone is placed at the corner of the cuboid, 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.2012-08-04
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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?2012-08-04
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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.2012-08-04
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    looks trivial to me!2012-08-05
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    The cylinder we're looking for has got to be at right angles to the corners, right?2012-08-10
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    The cylinder has radius x and height say r. There seems to me no assumption about orientation of the cylinder inside the box. In my opinion for small x it might be a bit difficut to calculate the value of r, (cylinder just fitting along diagonal) but the answer should be some function $f(x)$ which gives the max $r$ in all cases.2012-10-18

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