Cut Ellipsoid Volume

We start by transforming the given ellipsoid into the unit sphere (radius 1)

Define $x' = x / 10, y' = y / 15, z' = z / 30$ , then

$x'^2 + y'^2 + z'^2 = 1$

The corresponding cutting plane, is derived by substituting x, y, z into its equation

$(10 x') + 3(15 y') + 2(30 z') = 40$

or

$10 x' + 45 y' + 60 z' = 40$

which simplifies to

$2 x' + 9 y' + 12 z' = 8$

This plane cuts through the sphere. The distance from the center of the sphere is

$a = \dfrac{8}{\sqrt{2^2 + 9^2 + 12^2}} = \dfrac{8}{\sqrt{229}}$

Now we use the following formula for calculating the volume of a cut sphere of radius R, with

a plane that is a distance $a$ away from the center.

$V_S = \pi \left \{ \dfrac{2}{3} R^3 + a R^2 - \dfrac{a^3}{3} \right \}$

Substituting the calculated $a$ , and $R = 1$ , results in

$V_S = 1.14607276 \pi$

Finally we scale back this volume, to get the volume of the cut ellipsoid

$V = (10)(15)(30) V_S = 16202.222$ , therefore the answer is $\lfloor 16202.222 \rfloor = \boxed{16202}$ .