Volumes of aluminum cans

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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 $V$ , 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 $V = \pi \cdot r^2 \cdot h$ where $r$ is the radius of the cylinder and $h$ is the height of the cylinder. Also the surface area is given by $\begin{align*} A &= \underbrace{\pi \cdot r^2}_{\text{bottom}} + \underbrace{2\cdot\pi \cdot r\cdot h}_{\text{sides}} + \underbrace{\pi \cdot r^2}_{\text{top}}\\ &= 2\cdot \pi \cdot r^2 + 2\cdot\pi \cdot r\cdot h. \end{align*}$ 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 $V$ that uses the least amount of aluminum. In the context above, what do we want to minimize?

$A$ $V$ $h$ $r$

In the context above, what should be considered a constant?

$A$ $V$ $h$ $r$

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

Consider $r$ to be the variable, and express $A$ as a function of $r$ .

First, let’s eliminate the variable $h$ . We know that $V$ is a constant and that $V = \pi \cdot r^2 \cdot h$ so $h=\answer {V/(\pi r^2)}$ . Substitute this expression for $h$ in the equation $A = 2\cdot \pi \cdot r^2 + 2\cdot \pi \cdot r \cdot h.$

$A = \answer {2\cdot \pi \cdot r^2 + 2\cdot \pi \cdot r\cdot V/(\pi r^2)}$

Now consider $h$ to be the variable, and express $A$ as a function of $h$ .

First, let’s eliminate the variable $r$ . We know that $V$ is a constant and that $V = \pi \cdot r^2 \cdot h$ so $r=\answer {\sqrt {V/(\pi h)}}$ . Substitute this expression for $r$ in the equation $A = 2\cdot \pi \cdot r^2 + 2\cdot \pi \cdot r \cdot h.$

$A = \answer {2\cdot \pi \cdot V/(\pi h) + 2\cdot \pi \cdot \sqrt {V/(\pi h)} \cdot h}$

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.