This only requires single variable calculus

It was known to Archimedes that the volume of a paraboloid is half the volume of the circumscribed cylinder. Thus $\pi(x^2+y^2)z=\pi z^2=20$ and $z=\sqrt{\frac{20}{\pi}}\approx 2.5231$ . The answer is $\boxed{2523}$ .

You can quickly realize that this paraboloid is merely finding the volume of revolution of a parabola. We could put this as $y =x^2$ revolved around the y-axis.

The formula for such a problem is $\displaystyle \int_{0}^{D} \pi r^{2} \, dy$ . D denotes the depth we are looking for. What is the radius $r$ ? Remember, finding volumes of revolution is like finding the sum of volumes of infinite cylinders. Well, we are given $y = x^2$ and we need $x$ in terms of $y$ , so we could say $x = \sqrt{y}$ (it doesn't matter if you use the negative or positive version. I'm going to use positive version). Here, the $x$ value always happens to be our radius. Also, why $dy$ ? That would be the height of each cylinder, and this height lies on the y-axis.

The formula becomes:

$\displaystyle \int_{0}^{D} \pi y \, dy$

$= \displaystyle \left. \pi \dfrac{y^2}{2} \right|_{0}^{D} = \pi\dfrac{D^2}{2}$

Since the volume is 10, to solve for depth, set 10 equal to the equation we got.

$10 = \pi\dfrac{D^2}{2} \to D = \sqrt{\dfrac{20}{\pi}}$

$\left \lfloor 10^3 \sqrt{\dfrac{20}{\pi}} \right \rfloor = \boxed{2523}$