Two young mathematicians discuss optimizing aluminum cans.

Check out this dialogue between two calculus students (based on a true story):
Devyn
Riley, have you ever noticed aluminum cans?
Riley
So very recyclable!
Devyn
I know! But I’ve also noticed that there are some that are short and fat, and others that are tall and skinny, and yet they can still have the same volume!
Riley
So very observant!
Devyn
This got me wondering, if we want to make a can with volume , what shape of can uses the least aluminum?
Riley
Ah! This sounds like a job for calculus! The volume of a cylindrical can is given by where is the radius of the cylinder and is the height of the cylinder. Also the surface area is given by

Somehow we want to minimize the surface area, because that’s the amount of aluminum used, but we also want to keep the volume constant.

Devyn
Whoa, we have way too many letters here.
Riley
Yeah, somehow we need only one variable. Yikes. Too many letters.
Suppose we wish to construct an aluminum can with volume that uses the least amount of aluminum. In the context above, what do we want to minimize?
In the context above, what should be considered a constant?

As Devyn and Riley noticed, when we work out this type of problem, we need to reduce the problem to a single variable.

Consider to be the variable, and express as a function of .
First, let’s eliminate the variable . We know that is a constant and that so . Substitute this expression for in the equation
Now consider to be the variable, and express as a function of .
First, let’s eliminate the variable . We know that is a constant and that so . Substitute this expression for in the equation

Notice that we’ve reduced (one way or another) this function of two variables to a function of one variable. This process will be a key step in nearly every problem in this next section.