# Homework Help: Relationship between a given volume for a cylinder and the minimum surface area.

1. Apr 19, 2010

### stevie :)

I currently have a question that i am struggling with it is:

Propose a mathematical model in the form of an equation desribing, in general terms, the relationship between a given volume for a cylindrical container and the minimum surface area of material required to make it..

i am struggling with understanding this and actually defining an equation. please some assistance with this topic?????

2. Apr 19, 2010

You could write down the expressions for the volume V and the area A (both dependent on r and h), and work out V / A.

3. Apr 19, 2010

Edit: so you can get an equation of the form V = k A, where k is some coefficient dependent on r and h, and thus you have a relation V = V(A) which expresses the volume in terms of the area.

4. Apr 19, 2010

### stevie :)

ok just with that 1st comment i dont really understand that... could you show me how to work that out using r and h

5. Apr 19, 2010

If I got your question right, it doesn't really make any sense. Let's say you have a given volume of a cylinder, V. For this volume the surface (i.e. the material surface of the cylinder) is unique. What I meant (if this was your question) was to find how many surface of material you need for a certain volume, i.e. a relation between the volume and material needed to build it.

6. Apr 19, 2010

### HallsofIvy

That's not true. Both Volume and surface area of a cylinder depend on the height and radius: $V= \pi r^2 h$ and $S= 2\pi rh+ 2\pi r^2$. There will be different values of r and h which give the same volume but different surface areas.

With fixed V, $h= V/(\pi r^2)$ so the surface area is $S= (2\pi r)(V/(\pi r^2))+ 2\pi r^2= 2V/r+ 2\pi r^2$. Differentiate that with respect to r and set the derivative equal to 0 to find the minimum surface are for a given volume.

(Since this whole thread has nothing to do with "differential equations", I am moving it to "Calculus".)

7. Apr 19, 2010

### stevie :)

ok sorry bout putting in the wrong place...

i think what the question is asking is for any volume, like a general relationship that works for any volume and will give the minimum surface area for that volume...

8. Apr 19, 2010

### stevie :)

but by differentiating A = 2V/r+ 2\pi r^2 with respect to r doesnt that simply give you a value for a minimum radius of the container after you set it = 0????

9. Apr 19, 2010

### stevie :)

volume and surface area relationship

1. The problem statement, all variables and given/known data

Propose a mathematical model in the form of an equation desribing, in general terms, the relationship between a given volume for a cylindrical container and the minimum surface area of material required to make it..

2. Relevant equations

v=pi*r^2*h
a=2*pi*r*h + 2*pi*r^2

3. The attempt at a solution\

we are asked to find a general equation that works for any given volume and will define the minimum surface area for that volume...

no attempt as yet

10. Apr 19, 2010

### Staff: Mentor

Stevie -- you need to try harder. We do not do your homework or schoolwork for you here. We are here to help as you work through the problem.

11. Apr 20, 2010

### stevie :)

that's what im asking for... i dont understand how to find the relationship between any given volume and the minimum surface area so i'm asking for some assistance to get it started

12. Apr 20, 2010