I need to find out a way or strategy to draw a cylinder and two circles on paper but minimise the papers waste

Question

1. How to minimise the waste of paper when drawing the rectangle and two circles on paper which make up a cylinder.

2. R there any formulas or strategies for this

Please help!!! thanx

Answer 1

You need to fit two circles of diameter d and a rectangle of πd x h onto your paper. The area of paper used for the cylinder is d^2 + πdh. d + πd = W and h = 2d is one possibility. The other is d + πd = L. In the first case the waste will be
2d^2(1 - π/4) + (L - 2d)W
d = W/(1 + π)
2(W/(1 + π))^2(1 - π/4) + (L - 2W/(1 + π))W =
2W^2(1 - π/4)/(1 + π)^2 + LW - 2W^2/(1 + π) =
2W^2(1 - π/4)/(1 + π)^2 + LW - 2W^2(1 + π) /(1 + π)^2 =
LW + 2W^2[(1 - π/4) - (1 + π)] /(1 + π)^2 =
LW - (5π/2)W^2 /(1 + π)^2
and in the second case the waste will be
2d^2(1 - π/4) + (W - 2d)L =
LW - (5π/2)L^2 /(1 + π)^2
Since L > W (by definition)
Waste(min) = LW - (5π/2)L^2 /(1 + π)^2
and
d = L/(1 + π)
h = 2L/(1 + π)