2023 Cayley Problem 16

In the diagram, ABCF is a rectangle with AB = 30 cm and AF = 14 cm. Points E and D are on FC so that FE = 5 cm and the area of quadrilateral ABDE is 266 cm^2. The length of DC is

\textbf{(A)}\ 17\ \text{cm}\quad \textbf{(B)}\ 16\ \text{cm}\quad \textbf{(C)}\ 19\ \text{cm}\quad \textbf{(D)}\ 20\ \text{cm}\quad \textbf{(E)}\ 18\ \text{cm}

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.