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## Homework Statement

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 below.

[PLAIN]https://webwork.math.nau.edu/webwork2_course_files/JLevy_137/tmp/gif/08_Webwork-prob6-q35fig.gif [Broken]

Every cubic meter of water weighs 10000 newtons. What is the maximum possible weight for your boat and cargo?

I really just don't know what to do with this problem, although I set up what I thought was correct.

## Homework Equations

[itex]

Volume (V) = 2 L \int^{\sqrt(\frac{h}{a})}_0 [h - ax^2] dx

[/itex]

## The Attempt at a Solution

[itex]

V = 2 L \int^{\sqrt(\frac{h}{a})}_0 [h - ax^2] dx

[/itex]

[itex]

V = 2 L [ hx - \frac{ax^3}{3} ]^{\sqrt(\frac{h}{a})}_0

[/itex]

The boat can be theoretically infinite in size if there's a big enough ocean to put it in, right?

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