How to perform the efficient cutting on paper to http://nrich.maths.org/2664
I am trying to solve the problem presented in this link for a paper of any general integral size .But ,I am unable to come up with a specific formula or approach to solve this question.How should I make the most efficient cutting to obtain the cylinder having maximum volume?
 A: As MvG says, there are four configurations to consider.  I'll do one, the blue one in the page you link to.  Let the paper be $L$ long (horizontal) by $W$ wide (vertical).  If the radius of the can is $r$ and the height is $h$ we have $2 \pi r \le L, h+2r \le W, V=\pi r^2 h$.  One of the inequalities must be an equality or we can make the can larger.
Case 1: $2 \pi r = L$  Then $r=\frac L{2\pi}, h=W-2r, V=\pi\frac {L^2}{4\pi^2}(W-2r)=\frac{L^2}{4\pi}(W-\frac L\pi)$
Case 2: $h+2r=W$  Then $h=W-2r, V=\pi r^2(W-2r),\frac {dV}{dr}=2\pi r(W-2r)-2\pi r^2$  Setting the derivative to zero, we find $r=\frac W3, V=\frac {\pi W^3}{27}$
Case 2 works as long as $L \ge \frac {2 \pi}3 W$ and always beats Case 1 unless the paper is too short.
The others can be done similarly, then compared for a given $\frac LW$ ratio to determine the optimum.
A: There are basically $2\times2=4$ cases to distinguish. The rectangle which forms the side of the cylinder has two pairs of edges, one where it is glued to itself and one where it is glued to the caps. One of them will be parallel to the long side of the paper, and one to the short. One of them will be touching the cap circles and the other will not. For each of the resulting cases, you can compute the required size of the paper for given radius $r$ and height $h$ of the cylinder.
Then you can turn this around: given the size of the paper, you can use these equations to determine limits for $r$ and $h$. In some cases you might be able to choose these independently, and maximize each. In other cases, $r$ and $h$ will depend on one another, so you can express one in terms of the other and then express the volume as a function of a single parameter. Differentiating that function you can obtain a formula for the optimum. In the end you should have four volumes, each representing the maximum for a given case. Choosing among them should be easy, but might depend on the aspect ratio of the paper.
If you need more detailed answers, I would advise you to either update your question with the formulas you have found, or ask a new and more specific question regarding one step of the process.
A: Make the circles as big as you can make them, about half of the paper should be devoted to the circles. The other half should be for the rectangle, or middle part. 
Source: I am a 7th advanced math student doing this same problem. 
