When solving problems in Engineering School (Cal Poly) involving a lagoon in this shape I was told to approximate the volume with a trapezoidal prism. Although the shape is trapezoidal it is not a prism, so this always bothered me a bit. After using two methods (1) breaking the shape down into simpler polyhedra with known volume formulas such as prisms and pyramids and (2) using calculus to integrate the cross-sectional area as a function of height I have the “gold bar formula.”

V = (1/6)(2LW+2lw+Lw+lW)h

Where L and W are the length and width of the large base, l and w are the length and width of the small base and h is the height from base to base. Notice that in the case of a rectangular prism (L=l and W=w) the formula reduces to V=lwh, and in the case of a pyramid (l = w = 0) the formula reduces to V=(1/3)LWh.

This formula has applications not only for man made water bodies, and gold bars, but chocolate, pottery, and decorative containers.

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December 17, 2007 at 6:17 am

Finally! A mystery of the ages has been solved 🙂

July 19, 2008 at 6:51 pm

i like them all

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September 10, 2008 at 6:14 pm

Or, we could just make a model of it, and see how much water is displaced! The method looks good, and it is useful to understand; but one thing I’ve learned is that sometimes we overdo complexities when simple solutions abound – although, this is not always the case, so it’s helpful to know the math. One can modify the variables in the formula to suit the need, but models take longer to change. 😉 I still like the calculus methods for volume – it just seems neater to “slice” and “stack” the object.

September 14, 2008 at 1:51 am

So true, Chuck. I was explaining to my students how you could use a quadratic equation to describe the motion of a basketball and precisely make your shot. However, it would be near impossible to get the initial velocities right, so you might as well just forget about the science and learn the art of basketball.

I would like to build those water displacement models and maybe measure some everyday gold bar shaped objects like say…a gold bar, just to test these formulas out. If you’re up for the challenge I’d love to see the results.

October 3, 2008 at 3:03 pm

I say that calculas, yet i have not seen it, will be easy iin my senior year. I also say that this sight is boring.

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