4. The body diagonal of a rectangular prism is 10, and the projection of this diagonal on one of the surfaces of the prism is 8. Then the maximum volume of this rectangular prism is $\qquad$ .
4. 192 .

According to the problem, the height of the cuboid is $\sqrt{10^{2}-8^{2}}=6$. Let the side lengths of the base of the cuboid be $a$ and $b$. Then $a^{2}+b^{2}=64$.
Thus, the volume of the cuboid is $V=6 a b \leqslant 3\left(a^{2}+b^{2}\right)=192$.
The equality holds if and only if $a=b=4 \sqrt{2}$. Therefore, $V_{\max }=192$.