Calculus
posted by Sam .
What is the simplest solution to the Brachistochrone problem and the Tautochrone problem involving calculus? (I know that the cycloid is the solution but I need a simple calculus proof as to why this is the case)

The Brachistochrone (shortest sliding time) and the Tautochrone (equal sliding time wherever release occurs) are the same curve. Both are cycloids  loci of the a point at the edge of a wheel as it rolls. The situation is discussed in G. B. Thomas' classic textbook "Calculus and Analytic Geometry", 3rd edition, published by Addison Wesley in 1960. There is a ninth edition with the same title with Thomas and Finney listed as authors, published in 1999. I don't know if the proof is there. For a complete proof, Thomas refers to the book "Calculus of Variations" by G.A. Bliss, published in 1925. The original proof was by James and John Bernoulli.

a can containing 54 in cubed of tuna and water is to be made in the form of a circular cylinder. What dimensions of the can will require the least amount of material?