Task 178

The task

A rectangular box is 4 cm thick, and its square bases measure 16 cm by 16 cm. What is the distance, in centimeters, from the center point $P$ of one square base to corner $Q$ of the opposite base? Express your answer in simplest terms. [asy] import three; draw((0,0,1/4)--(1,0,1/4)--(1,1,1/4)--(0,1,1/4)--(0,0,1/4)--cycle,linewidth(2)); draw((0,1,0)--(1,1,0),linewidth(2)); draw((1,1,0)--(1,0,0),linewidth(2)); draw((0,1,0)--(0,1,1/4),linewidth(2)); draw((1,1,0)--(1,1,1/4),linewidth(2)); draw((1,0,0)--(1,0,1/4),linewidth(2)); dot((1/2,1/2,1/4)); dot((0,1,0)); label("$P$",(1/2,1/2,1/4),W); label("$Q$",(0,1,0),E); [/asy]