Answer
The cube's density as measured by an experimenter is $~~10,520~kg/m^3$
Work Step by Step
Let $L_0$ be the rest length of each side of the cube.
We can find the length $L$ of one of the sides according to the lab reference frame:
$L = L_0~\sqrt{1-\frac{v^2}{c^2}}$
$L = L_0~\sqrt{1-\frac{(0.9~c)^2}{c^2}}$
$L = L_0~\sqrt{1-0.81}$
$L =0.436~L_0$
Let $V_0$ be the volume of the cube at rest.
We can find the new volume in the lab reference frame:
$V = L\times L_0\times L_0$
$V = 0.436~L_0\times L_0\times L_0$
$V = 0.436~L_0^3$
$V = 0.436~V_0$
Let $m_0$ be the rest mass.
We can find the mass $m$ in the lab reference frame:
$m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$
$m = \frac{m_0}{\sqrt{1-\frac{(0.9~c)^2}{c^2}}}$
$m = \frac{m_0}{\sqrt{1-0.81}}$
$m = 2.294~m_0$
We can find the new density in the lab reference frame:
$\rho = \frac{m}{V}$
$\rho = \frac{2.294~m_0}{0.436~V_0}$
$\rho = 5.26~\frac{m_0}{V_0}$
$\rho = (5.26)~(2000~kg/m^3)$
$\rho = 10,520~kg/m^3$
The cube's density as measured by an experimenter is $~~10,520~kg/m^3$
