As a confectioner, you have a fastidious customer who demands a very specific three-layer cake. Each layer is a right circular cylinder, of volume V = r²h. The layers could vary in radius and height, but the entire cake must fit inside a hollow glass cone whose height and radius are both one foot in length. The customer also demands that this be the largest cake, in terms of volume, with those conditions. To construct such a cake, you will need to find the radius of each of the three layers. Express the total volume as a function of the three radii. We're looking for critical points. Take the three partial derivatives and set them equal to zero. This gives you a system of three nonlinear equations in three variables. Do not attempt to solve this by hand! Instead, use a computer algebra system, such as WolframAlpha, to get the exact radii of the three layers. What is the exact volume of the largest cake possible that fits under this glass cone? (not necessarily to scale)