The paper has certain dimensions, height = 21, width = 30.
A cylinder has a certain h, with circles of radius r at the base. Thus, if the radius of the circles are r, the width of the paper (30) must equal:
4*r + h = 30
(4 radii from two circles plus the height of the cylinder = width of the paper)
However the height of the paper must be equal to the circumference of one of the circles (2*pi*r) in order to form a cylinder:
2*pi*r = 21
thus r = (21/2*pi) or approximately 3.34 (cm)
giving us the value for h = 30 - 4*(21/2*pi). or approximately 16.63 (cm)
Making a cylinder with bases of diameter 6.68 cm and a height of 16.63 cm. and a volume of 2*pi*r^2*h = 2*3.14159*3.34^2*16.63
= 1165.64 cm^3
Which is the answer to the problem.
=== Warning, thinking outside the box:
However one can argue that one could put the height of the cylinder along the height of the paper, in which case:
4*r + 2*pi*r = 30 and (edit: that is, 4 radii of two circles plus the circumference of the cylinder's base = width of the paper)
h = 21
Which gives us as solutions:
r = 30/(4+2*pi) = approximately 2.92 cm
h = 21 cm
And a volume of 2*pi*r^2*h = 1125.03 cm^3
So theoretically there are two solutions to the problem. (but in any case the Volume of the cylinder is FIXED, that is a certain value).
JJ +