The gallery floor and ceiling make up the half-ellipse in this question. The semi-major axis (denoted by a)=47.3/2=23.65ft.
We can represent the length of the semi-minor axis by b. The ellipse equations is:
x2/a2+y2/b2=1 (centre at (0,0)).
The foci lie on the semi-major axis at (-c,0) and (c,0). We can find c by using the property of an ellipse: the sum of the distances of any point to each of its foci is a constant.
(-a,0), (a,0) and (0,b) all lie on the half-ellipse so (0,b) is a distance √(b2+c2) from each focus, making the combined distance 2√(b2+c2). (a,0) is distance a-c from one focus and a+c from the other, so the sum is 2a. Therefore since the sum is constant for all points a=√(b2+c2), and c2=a2-b2.
c=20.3ft, a=23.65ft, b2=a2-c2=(23.65-20.3)(23.65+20.3)=3.35×43.95=147.2325ft2.
Therefore b=12.1339ft approx. This is the height of the room at its centre.