(Solved): 7. A grain silo is to be built by attaching a hemispherical roof and a flat floor onto a circular c ...

7. A grain silo is to be built by attaching a hemispherical roof and a flat floor onto a circular cylinder. Use Lagrange multipliers to show that for a total surface area $S$, the volume of the silo is maximized when the radius and height of the cylinder are equal.