The Best Cake

Optimization is one of the most useful tools in all of math. Many of the visualizations here based in finding the optimum solution to something (and indeed, it drives many of the solution methods of differential equations). Here's an optimization puzzler brought to you by FiveThirtyEight. What are the optimal thicknesses of a 3-layer cake (each layer is cylindrical) that can fit into a given (right, circular) cone, that maximizes the volume? My solution is complicated, but here's a picture of the result... (it's red velvet)