Relationship between a given volume for a cylinder and the minimum surface area.

[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?

 

[radou]

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.

 

[radou]

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.

 

[stevie :)]

ok just with that 1st comment i don't really understand that... could you show me how to work that out using r and h

 

[radou]

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.

 

[HallsofIvy]

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.

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".)

 

[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...

 

[stevie :)]

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".)

but by differentiating A = 2V/r+ 2\pi r^2 with respect to r doesn't that simply give you a value for a minimum radius of the container after you set it = 0?

 

[stevie :)]

volume and surface area relationship

Homework Statement

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..

Homework Equations

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

The Attempt at a Solution

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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

 

[berkeman]

(two threads merged)

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.

 

[stevie :)]

that's what I am asking for... i don't 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

 

[radou]

(Halls, thanks - my apologies to Stevie for eventual misguide)