Lake Volume

Calculus Level 3

The height profile of a valley basin can be described by the two-dimensional parabolic functionh(x,y)=x2144m+y2324m. 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 h0=12m.h_0 = 12\,\text{m}. What is the volume of the resulting lake ((in m3)\text{m}^3) to the nearest integer?

Hint: Find the shape of the cross-sectional areas enclosed by the equipotential lines h(x,y)=z=constanth(x, y) = z = \text{constant}. The volume then results from the integral V=0h0A(z)dz\displaystyle V = \int_0^{h_0} A (z)\, dz over the cross-sectional area A(z).A(z).

×

Problem Loading...

Note Loading...

Set Loading...