Two young mathematicians discuss optimizing aluminum cans.
- Riley, have you ever noticed aluminum cans?
- So very recyclable!
- 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!
- So very observant!
- This got me wondering, if we want to make a can with volume , what shape of can uses the least aluminum?
- 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.
- Whoa, we have way too many letters here.
- Yeah, somehow we need only one variable. Yikes. Too many letters.
As Devyn and Riley noticed, when we work out this type of problem, we need to reduce the problem to a single variable.
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.