right circular cylindrical tin cans are to be manufactured to contain 400 cm ^3 of volume. There is no waste involved in cutting the tin that goes into the vertical sides of the can, but each end piece is to be cut from a square and the corners of the square wasted. Find the ratio of the height to diameter for the most economical cans.

Question

Reiny · Answer

I will assume that by "most economical" you mean
"minimum cost of material for surface area".

let each side of the end square be x, so the radius for the end circles is also x
let the height be y
SA = 2x^2 + 2πxh
but V = 400
400 = πr^2h
h = 400/(πr^2)

SA = 2x^2 + 2πx(400/(πx^2)
= 2x^2 + 800/x

find d(SA)/dx
set it equal to zero and solve for x
straightforward from here.