Question

You are asked to build an open cylindrical can (i.e. no top) that will hold 364.5 cubic inches. To do this, you will cut its bottom from a square of metal and form its curved side by bending a rectangular sheet of metal.
Express the total amount of material required for the square and the rectangle in terms of r.

Answer 1

Generate a function for the height of the cyliner as a function of radius. Use this function to generate a function of r for the material used. Take the derivative of that function, set it to zero, and solve for r.

V=πr2hh=d h=d=2r h=2r V=πr22r

V(r,h)=180.5=πr2h⟹h=180.5/πr2⟹A(r)=r(r+h)=r2+180.5/πr

A(r)=2r(2r+h)=4r2+361/πr

A′(r)=8r−361/πr2