An open-top rectangular tank with a square base is to be constructed to hold exactly 2,000 cm3 of liquid. The base must be reinforced, doubling the material used for that face only, while the four vertical sides each require a single layer of material. Let x cm be the length of one side of the square base and let y cm be the height of the tank.
(a) Express the total surface area S(x,y) of material that will be used, taking into account the doubled material on the base.
(b) Use the volume constraint to write y as a function of x and rewrite S as a function S(x) of one variable.
(c) Find the value of x that minimizes the total surface area of material, and determine the corresponding height y.
(d) Justify that the dimensions you obtained indeed minimize the surface area.
(e) Compute the minimum total amount of material required, stating appropriate units.