You are asked to build an open cylindrical can (i. e. no top) that will hold 171.5 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.

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=π r2 h h=d h=d=2r h=2r V=π r2 2r

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

A (r) = 2r (2r+h) = 4 r2 + 361 / πr

A′ (r) = 8r - 361 / π r2