In this project, you play the role of a consultant to a bedding manufacturer. The manufacturer is interested in constructing a sleeping pod. You will use a number of tools from Calculus III (including optimization and parameterization) to advise the client on the construction of such a pod. The pod is to have an outer shell in the shape of the ellipsoid 3x + y + 1 z = 3. 1. Use a graphing utility to graph the ellipsoid in 3-space. 4 2. Derive a parameterization of the ellipsoid in terms of p and 0 (in spherical coordinates). 3. Find the volume of the ellipsoid. You need to exhibit the appropriate integration to find this volume. 4. The pod is to have walls that are uniformly 0.15 units thick. The construction material for the walls costs $1.17 per cubic unit. What is the cost to produce the walls of the pod? 5. Find the point(s) on the inner shell of the pod that is farthest from the origin and such that x = 2. 6. Using Lagrange Multipliers, find the (right) rectangular prism of maximum volume that can be inscribed in the inner shell of the pod. Assume that the faces of the prism are parallel to the coordinate planes. Describe the prism algebraically and/or completely describe its eight vertices.