Maximal volume of cup for a given area of material

[helpm3pl3ase]

Perfect Paper Cup Company (PPCC) wants to make a cylindrical cup (with
open top, of course, so people can drink out of it). The cup is to be made from
exactly 22 in2 of paper—not counting any paper wasted in cutting out the circular
bottom, in attaching the bottom to the cylindrical piece, or in fastening
together the ends of the rectangle that’s rolled up to make the cylindrical
piece.

I don't know hwere to beginnnnn

 

[matt grime]

Given an open cyclinder of radius R and height H, what are the equations of the volume and surface area?

 

[helpm3pl3ase]

hmmm did i start this correctly??

A = 2(pi)r^2 + 2(pi)rh

(pi)r^2h = 22

h = 22/(pi)r^2 because h is what we want to maximize??

 

[matt grime]

I'm going to play devil's advocate. What is A?

You didn't actually say what the question asked you to maximize. One presumes it is the volume normally.

 

[Jheriko]

hmmm did i start this correctly??

A = 2(pi)r^2 + 2(pi)rh

(pi)r^2h = 22

If the cup is a cylinder with an open end then shouldn't this be $A = \pi r^2 + 2 \pi r h$, so that $\pi r^2 + 2 \pi r h = 22$? The volume is $V = \pi r^{2} h$.

The problem might become easier if you use the expression for the surface area to eliminate either r or h from the equation for the volume.