The Problems 1. Designing the Best Gutter: Your friend Eric works for a company in Seattle that produces rain gutters. A natural design is to start with a long flat sheet of galvanized steel, and then bend the sides up as shown below. The sheet has a constant width w. Eric has been tasked with optimizing this design, and would like your help. w-2x 0 (a) Eric wants to know the values of x and that allow the greatest flow; that is, that give the maximum possible cross-sectional area. Find them. (b) The company is also considering a new design, where the same sheet is bent into a semicircle. Would this be better? 2. Furthest Tangent Plane: Consider the surface x3yz = 1. Find the tangent plane to this surface that is furthest from the origin. Note this is not the tangent plane to the point on the surface furthest from the origin, since the surface extends arbitrarily far from the origin! Rather, you are looking for the plane tangent to the surface whose minimum distance from the origin is largest. Here's some guidance to help you: Find the equation for the tangent plane to the surface at an arbitrary point. What is the minimum distance from this tangent plane to the origin? Assuming you have a point on the surface xyz 1, try to simplify this distance to a function of two variables. Maximize this distance function, and write down the equation for the desired plane.