2015 Cayley Problem 25

Rectangular prism PQRSWTUV has a square base PQRS. Point X is on the face TUVW so that PX = 12, QX = 10 and RX = 8. The maximum possible area of rectangle PQUT is closest to

\textbf{(A)}\ 67.84\quad \textbf{(B)}\ 67.82\quad \textbf{(C)}\ 67.90\quad \textbf{(D)}\ 67.86\quad \textbf{(E)}\ 67.88

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.