SOLUTION: The opening of a cave 20 feet wide is in the shape of a semi ellipse. If the height of the opening is 15 feet at the center, how high is the opening 5 feet from the center? Pls

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: The opening of a cave 20 feet wide is in the shape of a semi ellipse. If the height of the opening is 15 feet at the center, how high is the opening 5 feet from the center? Pls       Log On


   

Question 970979: The opening of a cave 20 feet wide is in the shape of a semi ellipse. If the height of the opening is 15 feet at the center, how high is the opening 5 feet from the center?
Pls Help me.
Answer by mathmate(429) About Me  (Show Source):
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The opening of a cave 20 feet wide is in the shape of a semi ellipse.
If the height of the opening is 15 feet at the center, how high is the opening 5 feet from the center?

Given:
A semi-ellipse opening 20' wide and 15' high at the centre.
Find height of opening at 5' from centre.

Solution:
We have an ellipse centred at origin, with
a = semi-x-axis = 20%2F2 = 10'
b = semi-y-xaxis = 15'

The standard equation of an ellipse is
%28x%2Fa%29%5E2%2B%28y%2Fb%29%5E2+=+1
Substituting a and b,
%28x%2F10%29%5E2%2B%28y%2F15%29%5E2+=+1

At 5' from centre, x=5:
%285%2F10%29%5E2%2B%28y%2F15%29%5E2+=+1

Solve for y:
%28y%2F15%29%5E2+=+1-%285%2F10%29%5E2+=+3%2F4
y=15+%28sqrt%283%2F4%29%29+=+12.99' approximately

Answer:
The height of the cave opening at 5' from the centre is approximately 12.99'