Determine which shape minimises the volume of the material used to build a drain made by a fixed length of material, l, fixed dimension of material, x by y, and wetted perimeter must be of, w.

Note that x = 50cm, y = 100cm.

Generalise each shape with a sensible value for fixed length, l and fixed wetted perimeter, w respectively. By using calculus, determine the dimension of each shape and associated cross-sectional area that minimises the volume of the material used to build a drain.

b C X W W W y