Question

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)

Answer 1

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.

Answer 2

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?