The design of boats is based on Archimedes' Principle, which states that the buoyant force on an object in water is equal to the weight of the water displaced. Suppose you want to build a sailboat whose hull is parabolic with cross-section y=ax^2, where a is a constant. Your boat will have length L and its maximum draft (the maximum vertical depth of any point of the boat beneath the waterline) will be H. See the figure below. Every cubic meter of water weighs 10^4 newtons. What is the maximum possible weight for your boat and cargo?

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