Lake Volume

Calculus Level 2

The height profile of a valley basin can be described by the two-dimensional parabolic function\[ h(x, y) = \frac{x^2}{144\,\text{m}} + \frac{y^2}{324\,\text{m}}. \] Now the basin is filled by a rainstorm to a height of \(h_0 = 12\,\text{m}.\) What is the volume of the resulting lake \((\)in \(\text{m}^3)\) to the nearest integer?

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Hint: Find the shape of the cross-sectional areas enclosed by the equipotential lines \(h(x, y) = z = \text{constant}\). The volume then results from the integral \(\displaystyle V = \int_0^{h_0} A (z)\, dz\) over the cross-sectional area \(A(z).\)

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