Question

The Whispering Gallery in the Museum of Science and Industry in Chicago is 47.3 feet long and is in the shape of an ellipse. The distance from the center of the room to the foci is 20.3 feet. Find an equation that describes the shape of the room. How high is the room at its center? 

Answer 1

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.