Using LaGrange Multipliers, solve the following problem. It is desired to build a rectangular box with a square base and without a lid, which has a volume of 32 ft3. What would be the least amount of material needed for its construction?

We wish to empty a hemispherical tank full of water out the top. Which should entail the larger amount of work: when the tank is like a cup full of water, or when the tank is upside down on a flat surface, with a hole at the top of the tank? Explain your answer, and include a picture of the situation.

A cylinder shaped can needs to be constructed to hold 250 cubic centimeters of soup. The material for the sides of the can costs 0.02 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.08 cents per square centimeter. Find the dimensions for the can that will minimize production cost. Helpful information: h: height of can, r: radius of can Volume of a cylinder: V = T Area of the sides: A = 2arh Area of the top/bottom: A To minimize the cost of the can: Radius of the can: Height of the can: Minimum cost: cents