A piece of stained glass art is shaped like two mountains, as shown in the figure below. One mountain peak is $6$ feet high while the other peak is $10$ 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 $120$ 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 $h$?