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.

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=πr2h h=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 / π r2