# Trouble with minimum surface area for a cylinder

• rela
In summary, the minimum surface area of a cylinder can be derived by taking the first derivative of the area equation, setting it equal to 0, and solving for the variable that is not being differentiated. This is done by fixing one of the variables (either r or h) as a constant, usually the volume. While this may seem unfair, it is simply a mathematical approach to finding the minimum surface area.
rela
Dear all,

I was reading through my notes and I was kinda like stumbled in the way the minimum surface area of a cylinder has been derived.

First,

A= 2*PI*r^2 + 2*PI*r*h

and given the condition that the volume has been fixed, the resulting area equation becomes

A= 2*PI*r^2 + 2V/r since V = PI*r^2*h

Taking the first derivative of A,

A' = 4*PI*r - 2V/r^2

Letting A' = 0 and solving for h will give us h = 2r.

Everything seems nice and well defined. However, I have some confusion in my head.

In the first place why can't we take the derivative of the first area equation directly (A= 2*PI*r^2 + 2*PI*r*h) and fixing h to be like a constant? I tried doing it and i got h = -2r? The negative sign simply indicates that it's not right.. I'm puzzled..?

And also,since volume is also a function of both 'r' and 'h', it doesn't really make sense to fix volume as a constant. I just find it weird. I mean we can fix 'h' for the volume but 'r' is still in it which in fact causes the volume to vary still. So how in the first place can we fix the volume with 'r', a variable being in it the first plcae?

Gosh, Hope I'm sounding right. I look foward to valuable inputs from all of you.

Thanks
Rela

rela said:
In the first place why can't we take the derivative of the first area equation directly (A= 2*PI*r^2 + 2*PI*r*h) and fixing h to be like a constant? I tried doing it and i got h = -2r? The negative sign simply indicates that it's not right.. I'm puzzled..?

Because you have to vary r and h at the same time (because they're connected - increase one, it automatically decreases the other).

But you can't differentiate with respect to both, so you eliminate one of them (either h or r - try it with r instead, just for practice!) by expressing in terms of the other and V.

Since V is a constant, you now have only one variable, and you can go ahead.

And also,since volume is also a function of both 'r' and 'h', it doesn't really make sense to fix volume as a constant. I just find it weird. I mean we can fix 'h' for the volume but 'r' is still in it which in fact causes the volume to vary still. So how in the first place can we fix the volume with 'r', a variable being in it the first plcae?

As a matter of "natural justice", yes, there's nothing to choose between r h and V - they're all variables.

But we can make anything a constant - in this example, we've chosen V.

Unfair? Maybe. But it's just maths …

Blast, another person who is giving excellent responses before I get to them!

I particularly like the "natural justice"!

## 1. What is the minimum surface area for a cylinder?

The minimum surface area for a cylinder is when the cylinder is a perfect sphere, with a surface area of 4πr².

## 2. Why is minimum surface area important for a cylinder?

Minimum surface area is important for a cylinder because it is the most efficient shape for containing a given volume of material. This is especially important in industries where materials and resources are limited or costly.

## 3. How do you calculate the minimum surface area for a cylinder?

The minimum surface area for a cylinder can be calculated using the formula 4πr², where r is the radius of the cylinder. This formula can also be used to calculate the minimum surface area for any spherical shape.

## 4. What factors affect the minimum surface area for a cylinder?

The minimum surface area for a cylinder is affected by the radius of the cylinder, as well as the material used to create the cylinder. The smoother the surface of the cylinder, the closer it will be to a perfect sphere and the lower the minimum surface area will be.

## 5. Can the minimum surface area for a cylinder be improved?

The minimum surface area for a cylinder is already the most efficient shape for containing a given volume of material. However, it can be improved by using materials with lower surface tension, such as liquids, which can conform to the shape of the container and reduce surface area even further.

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