The design of boats is based on Archimedes\' Principle, which states that the bu

Question

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 water line) will be H . See the figure. Every cubic meter of water weighs 10,000 newtons. What is the maximum possible weight for your boat and cargo

Explanation / Answer

y=ax^2

We fix the co-ordinate system at the bottom

H=ax^2

Therefore x = sqrt{H/a) and x=-sqrt{H/a}

Width of boat = 2*sqrt{H/a}

Therefore,

volume of boat = 2/3*H*L*2*sqrt{H/a}

Max weight of boat and cargo = Volume*10000 newtons

Assumption: H and L are in metres

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