I have a semi-project due tommorrow that basically asks the following question: If you are designing a tent in the shape of a spherical cap (a sphere with the lower portion sliced away by a plane) and the material used for the roof costs 2.5 times more per square foot than the material used for the floor, what should the dimensions of the tent be to minimize the cost of materials used? Additional info: the volume of the tent will be 150 cu ft. Goal: To derive a formula for maximum cost efficiency while retaining the original volume and shape of the tent. The most difficult part of this problem is simply knowing where to start. I realize that when optimizing, I have to keep the minimums and maximum values of my equation in mind. In this case, I want to use as little ceiling material as possible for the tent while making sure that it remains a dome-shaped tent. Formulas used in this problem are the "volume of a dome" equation which I believe is V=((pi*h)/6)(3r^2+h^2) and the surface area formula of a dome: S=pi(h^2+r^2). I wish I knew where to go from this beginning point, because short of taking the derivatives of these functions (with respect to what though?), I don't know to do.