Optimal Cake

Ollie Reeder at FiveThirtyEight poses this problem: (http://fivethirtyeight.com/features/can-you-bake-the-optimal-cake/) A mathematician who has a birthday coming up asks his students to make him a cake. He is very particular (he is a constructive set theorist, which explains a lot) and asks his students to make him a three-layer cake that fits under a hollow glass cone he has on his desk. (The cone was given to him as a prize for proving an obscure theorem long ago.) He requires that the cake fill up the maximum amount of volume under the cone as possible and that the layers of the cake have straight vertical sides. (Again, the guy’s particular.) What are the thicknesses of the three layers of the cake in terms of the height of the glass cone?