## Riddler Classic

This Sunday, March 14, is Pi Day! To celebrate, you are planning to bake a pie. You have a sheet of crust laid out in front of you. After baking, your pie crust will be a cylinder of uniform thickness (or rather, thinness) with delicious filling inside.

To maximize the volume of your pie, what fraction of your crust should you use to make the circular base (i.e., the bottom) of the pie?

## Solution

Assuming we use the entire sheet of crust without wasting to create the hollow cylinder with uniform thickness. We need to maximize the volume $\pi r^2 h$ subject to the constraint $2\pi r h + 2\pi r^2$ is constant.

We have the Lagrangian,

\begin{align} \mathcal{L}(r,h) = \pi r^2 h - \lambda(2\pi r h + 2\pi r^2 - k) \end{align}

Taking the partial derivatives of the Lagrangian w.r.t $h$ and $r$ and setting them to $0$, we have

\begin{align} \frac{\partial\mathcal{L}(r,h)}{\partial r} &=& 2\pi r h - \lambda(2\pi h + 4\pi r ) &=& 0 \\ \frac{\partial\mathcal{L}(r,h)}{\partial h} &=& \pi r^2 - 2 \lambda \pi r &=& 0 \end{align}

Solving the above simultaneous equations, we have

\begin{align} \lambda &=& \frac{r}{2} \\ h &=& 2r \end{align}

The fraction of the crust that should be used to make the circular base so as to maximize the volume of the pie is $\frac{\pi r^2}{2\pi r h + 2\pi r^2} = \frac{1}{6}$.