Word problem involving comic sections. A hall 100 feet in length is designed as a whispering gallery (See example 9 for Ellipses). If the focus is 25 feet from the center, how high will the ceiling be at the center?

Step 1 Given, The shape of whispering gallery is an ellipse of notation. Step 2 The center of gallery is ${\displaystyle (0,0).}$
${\displaystyle \text{Foci}=(\pm 25,0)}$
${\displaystyle \text{Vertices}=(\pm 50,0)}$ Step 3 So, ${\displaystyle a=50\text{and}c=25}$ Therefore, ${\displaystyle c^2=a^2-b^2}$
${\displaystyle {25}^2={50}^2-b^2}$
${\displaystyle b^2={50}^2-{25}^2}$
${\displaystyle b^2=2500-625}$
${\displaystyle b^2=1875}$
${\displaystyle b\approx 43.30}$ Step 4 Here b is length of minor axis, which is height of ceiling. So, the height of ceiling is equal to 43.3 feet.