Question 152620: Would someone kindly help me with this algebra question?
A satellite has an elliptical orbit around the earth with one focus at the earth’s center, E. The earth’s radius is 4,000 miles. The highest point that the satellite is from the surface of the earth is 800 miles, and the lowest is 200 miles. What is the eccentricity of the satellite’s orbit?
Thank you in advance for sharing your valuable knowledge :0) !!!
Grateful,
Jai
Found 2 solutions by scott8148, Edwin McCravy:
Answer by scott8148(6628)   (Show Source):
You can put this solution on YOUR website! thanks to a memory refresher from Wikipedia (a good resource)
the eccentricity is the difference of the orbital radii divided by the sum of the orbital radii
e=(4800-4200)/(4800+4200) __ e=600/9000 __ e=2/30 or .067 (approx)
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website!
Eccentricity = c/a
where c is the distance from the center to the focus of the ellipse
a is the distance from the center to a vertex
Here is a sketch:
On this graph 1 unit = 1000 miles. The circle represents
the earth with the center at the origin. so point P has
coordinates (4,0) and point Q has coordinates (-4,0)
The green ellipse represents the orbit of the satellite.
Where 1 unit = 1000 miles. So the satellite is 800 miles or 0.8 units
from the earth's surface at point M, and 200 miles or 0.2 units from
the earth's surface at point N. PM is 800 miles or 0.8 units and
QN is 200 miles or 0.2 units.
So the coordinates of M are (4.8,0)
and the coordinates of N are (-4.2,0)
Unfortunately the software does not close the green ellipse at its
vertices, but its vertices are at M and N. The center of the ellipse,
R, is the midpoint between M and N, so we use the midpoint formula
midpoint = ( ,  ),
midpoint = ( , ) = (0.3,0)
and find that the center of the ellipse is R(0.3,0)
Since the focus of the ellipse is at the center of the earth (0,0)
and the center of the ellipse is at R(0.3,0), the value of c
is c=0.3 units (distance from center to focus). That is c = 300 miles.
the value of a is a= (the distance from the ellipse's center
(.3,0) to vertex M(4.8,0) is 4.8-.3 or 4.5 units, or a = 4500 miles.
Therefore the eccentricity = 
Edwin
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