jmc

Suppose that the dimensions of the rectangular prism are given by $x$, $y$, and $z$, such that $xy=30,yz=180,$ and $zx=24$. If we multiply all three equations together, we get that $xy\cdot yz\cdot zx=(xyz{)}^2=30\cdot 180\cdot 24$. Using prime factorizations, we find that the right-hand side is equal to $(2\cdot 3\cdot 5)\times (2^2\cdot 3^2\cdot 5)\times (2^3\cdot 3)=2^6\cdot 3^4\cdot 5^2$. Thus, $(xyz{)}^2=(2^3\cdot 3^2\cdot 5{)}^2$, so $xyz=\boxed{360}$. This is the formula for the volume of the box.