jmc

By the definition of direct variation, we know that $x=m{y}^3$ for some constant $m$. By the definition of inverse proportion, we know that $y=n/\sqrt{z}$ for some constant $n$. Substituting for $y$ in the first expression, we can see that $x=\frac{m{n}^3}{(\sqrt{z}{)}^3}=\frac{k}{z\sqrt{z}}$ or $xz\sqrt{z}=k$ for some constant $k$. Substituting the given values, we can solve for $k$: $$xz\sqrt{z}=3\cdot 12\sqrt{12}=36\cdot 2\sqrt{3}=72\sqrt{3}=k$$Now, we can let $z=75$ and use the value of $k$ to solve for $x$: $$\begin{array}{rl}xz\sqrt{z}=x(75\sqrt{75})& =72\sqrt{3}\\ \Rightarrow x(75\cdot 5\sqrt{3})& =72\sqrt{3}\\ \Rightarrow 375\sqrt{3}x& =72\sqrt{3}\\ \Rightarrow x& =72/375=\boxed{\frac{24}{125}}\end{array}$$