Maximal volume of cup for a given area of material

In summary, the Perfect Paper Cup Company (PPCC) wants to make a cylindrical cup with an open top using exactly 22 in2 of paper. To maximize the volume, the equation should be A = πr^2 + 2πrh, with the volume V = πr^2h. The problem can be simplified by using the surface area equation to eliminate either r or h from the volume equation.
  • #1
helpm3pl3ase
79
0
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
 
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  • #2
Given an open cyclinder of radius R and height H, what are the equations of the volume and surface area?
 
  • #3
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??
 
  • #4
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.
 
  • #5
helpm3pl3ase said:
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 [itex]A = \pi r^2 + 2 \pi r h[/itex], so that [itex]\pi r^2 + 2 \pi r h = 22[/itex]? The volume is [itex]V = \pi r^{2} h[/itex].

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.
 

What is the formula for calculating the maximal volume of a cup for a given area of material?

The formula for calculating the maximal volume of a cup is V = (1/3)πr2h, where r is the radius of the cup and h is the height of the cup.

How do I determine the appropriate height and radius for the cup to achieve the maximum volume?

The appropriate height and radius for the cup can be determined by using the formula V = (1/3)πr2h. By substituting different values for r and h, you can calculate the volume and choose the combination that gives the maximum value.

What factors can affect the maximal volume of a cup for a given area of material?

The factors that can affect the maximal volume of a cup include the type of material used, the shape and design of the cup, and the accuracy of measurements. Additionally, the thickness of the material and any irregularities in the shape of the cup can also impact the maximal volume.

Can the maximal volume of a cup be greater than the given area of material?

No, the maximal volume of a cup cannot be greater than the given area of material. The area of material is a limiting factor, and the maximal volume can only be achieved by using the entire area of material to form the cup.

How is the maximal volume of a cup for a given area of material relevant in real-world applications?

The calculation of maximal volume of a cup for a given area of material is relevant in industries such as packaging and manufacturing. It helps in optimizing the use of materials and designing efficient products with maximum capacity. It also helps in reducing waste and costs associated with production.

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