Question 1151128
Rolling the cylinder lengthwise means that the circumference of the cylinder becomes the magnitude of the {{{width=W}}}, and the height is the magnitude of the {{{length=L}}}


so, if the circumference of the cylinder is {{{2pi*r = W}}}

{{{r = W/(2pi)}}}

Then the volume is:

{{{V[1] = pi*r^2*L}}}

{{{V[1] = pi(W/(2pi))^2*L}}}

{{{V [1]= pi(W^2/(4*pi^2))L}}}

{{{V[1] = (1/(4pi))W^2*L}}}



if the circumference of the cylinder is{{{2pi*r = L}}}


{{{r = L/(2pi)}}}

Then the volume is:

{{{V[2] = pi*r^2*W}}}

{{{V[2] = pi(L/(2pi))^2*W}}}

{{{V [2]= pi(L^2/(4*pi^2))W}}}

{{{V[2] = (1/(4pi))L^2*W}}}



Dividing the width folding by the length folding:


{{{(1/(4pi))W^2*L / (1/(4pi))L^2*W =cross((1/4pi))W^cross(2)*cross(L) / cross((1/4pi))L^cross(2)*cross(W) =W/L}}}

Thus the ratio of the two volumes is the ratio of the length to the width (interesting.) So the width folding has greater volume.