We have a rectangular sheet with length = 12 and width = 8.
The rectangle is rolled long-side to shape a cylinder.
Then the height of the cylinder will be the length of the rectangle= 12.
==> The height (h) = 12.
Then, the width of the rectangle will form the circumference of the base.
==> The circumference C = 8.
Now we will calculate the volume.
We know that the volume v = r^2 * pi * h where r is the radius of the base.
But we have C = 8
==> C = 2* pi * r = 8
==> r= 8/ 2*pi = 4/pi = 1.27 ( approx.)
Now we will substitute into the volume.
==> V = r^2 * pi * h
==> V = ( 4/pi)^2 * pi * 12
= 16/ pi * 12 = 61.12.
Then, the volume of the cylinder is 61.12 cubic units.
If you fold the sheet 8*12 along the longer side and join the 2 paralle widths, it becomes a cylinder of circumfernce 12 and height 8.
So the radius r of the cyliber whose circumference is 12 is given by :
r = circumference /2pi = 12/2pi = 6/pi.
The volume V of the cylinder whose radius is r is given by:
V =( pi*r^2)h, where h = is height. Here, r = 6/pi and h = 8
Therefore V = 2pi* (6/pi)^2 *8 = 576/pi = 183.35 cubic units.
Therefore the volume of the cylinder = 183.35 cubic units.