Question

Jamie and Jason want to exchange secrets (terrible secrets) from across a crowded whispering gallery. Recall that a whispering gallery is a room which, in cross section, is half of an ellipse. If the room is 15 feet high at the center and 60 feet wide at the floor, how far from the outer wall should each of them stand so that they will be positioned at the foci of the ellipse

Answer 1

4 feet

Explanation:

Since the whispering gallery is 15 feet high and 60 feet wide, it forms an ellipse with major axis along the width of the gallery and minor axis along the height of the gallery.

Using the equation of an ellipse with major axis along the x-axis, we have

x²/a² + y²/b² = 1 where a = midpoint of the width = 60/2 = 30 feet and b = 15 feet.

c² = a² - b² where c = focus of the ellipse.

So, c² = a² - b²

= 30² - 15²

= 900 - 225

= 675

c = ±√675 = ±25.98 feet

So, the coordinate of the focus is thus (0, ±c) = (0, ±25.98) and the coordinate of the vertex is (0, ±a) = (0, ±30)

So, the distance between the foci and the vertex is the distance from the wall at which each of them should be positioned at the foci.

So, d = a - c

= 30 ft - 25.98 ft

= 4.02 feet

≅ 4 feet