Calculating Volume of Cylinder w/ Fixed Dimensions on A4 Paper

[Clef]

Homework Statement

Explain why, using this outline for creating the cylinder, the volume of the cylinder is fixed for an A4 piece of paper (30x21)

Homework Equations

h=30-4R

I really don't know how to approach this question, i know it has something to do with the radius and the height of the cylinder. but i don't know how to prove it.

 

[Ks. Jan Jenkins]

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 +

 

[Ks. Jan Jenkins]

Bonus Question

Bonus Question !

Here is a bonus question that I thought of while looking at your question. It actually has an interesting answer:

http://ibphysics.org/images/cylinder-problem.jpg

 

[Clef]

The maximum volume is:


607.3 cm^3