2024 AMC 8 Problem 24

Jean 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 and the other peak is 12 feet high. Each peak forms a 90^\circ angle, and the straight sides of the mountains form 45^\circ angles with the ground. The artwork has an area of 183 square feet. The sides of the mountains 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)} 6 \qquad \textbf{(E)} 5\sqrt{2}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.