AUSTRIAN-POLISH 1997 Problem 9

Given a parallelepiped $P$ , let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$ , let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$ . Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$ .