jmc

If the length, width, and height of the rectangular prism are $a$, $b$, and $c$, then we are given $ab=48$, $bc=49$, and $ac=50$. Since we are looking for $abc$, the volume of the rectangular prism, we multiply these three equations to find $$\begin{array}{rl}(ab)(bc)(ac)& =48\cdot 49\cdot 50⟹\\ a^2{b}^2{c}^2& =48\cdot 49\cdot 50⟹\\ (abc{)}^2& =48\cdot 49\cdot 50⟹\\ abc& =\sqrt{48\cdot 49\cdot 50}\\ & =\sqrt{(16\cdot 3)\cdot 7^2\cdot (2\cdot 5^2)}\\ & =4\cdot 7\cdot 5\sqrt{2\cdot 3}\\ & =140\sqrt{6},\end{array}$$ which to the nearest whole number is $\boxed{343}$ cubic units.