Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4\sqrt{2} \qquad \textbf{(D)}\ 5\sqrt{2} \qquad \textbf{(E)}\ 6$
Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle as shown. 

The side length of the largest right triangle is $12\sqrt{2},$ which means its area is $144.$ Similarly, the area of the second largest right triangle is $64$ (the side length is $8\sqrt{2}$), and the area of the overlap is $h^2$ (the side length is $h\sqrt{2}$). Thus,
\[144+64-h^2=183,\]
which means that the answer is $\boxed{\mathbf{(B)}\text{ 5}}.$
~BS2012