SOLUTION: A whispering gallery has a semielliptical ceiling that is 9 m high and 30 m long. How high is the ceiling above the two foci? Also, find the distance of the focus from the center

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A whispering gallery has a semielliptical ceiling that is 9 m high and 30 m long. How high is the ceiling above the two foci? Also, find the distance of the focus from the center      Log On


   

Question 1167019: A whispering gallery has a semielliptical ceiling that is 9 m high and 30 m long. How high is the ceiling above the two foci? Also, find the distance of the focus from the center, equation of the ellipse, and answer to the problem.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The major axis is 30 feet and horizontal; the semi-minor axis is 9 feet and vertical. So the equation is

x%5E2%2F15%5E2%2By%5E2%2F9%5E2+=+1

a=15 is the semi-major axis; b=9 is the semi-minor axis; the distance from the center to each focus is c, where c%5E2+=+a%5E2-b%5E2

c%5E2=15%5E2-9%5E2+=+225-81+=+144
c+=+12

The height of the ceiling above each focus is the y value when x=c=12.

144%2F225%2By%5E2%2F81=1
y%5E2%2F81+=+81%2F225
y%5E2=81%5E2%2F15%5E2
y+=+81%2F15+=+27%2F5

ANSWERS:
The height of the ceiling above each focus is 27/5 = 5.4m.
Each focus is 12m from the center.
The equation of the ellipse is x%5E2%2F225%2By%5E2%2F81=1