Ellipse Word Problem. Need help!

[discreditedvalidity]

Here is the word problem:

A satellite is in an elliptical orbit around the earth with the center of the earth at one focus. The heigh of the satellite over the earth varies between 140 and 440 mi. Assume th earth is a sphere with radius 3960 mi. Find an equation for the path of the satellite with the origin at the center of the earth.


Okay, so far I have done this, and I don't know if I'm doing it right:

2a = 140 + 2(3960) + 144
= 140 + 7920 + 440
= 8500

a= 4250

a - c = 3960 + 140 = 4100
c = a- 4100
c = 4250 - 4100
= 150

foci = (+150, 0) (-150, 0)

c^2 = a^2 - b^2
b^2 = a^2 - c^2
= 4250^2 - 150^2
=18040000



That's all I have.
1) Have I done right so far? If not, what am I doing wrong?
2) How on earth do I get the equation?

Thanks!

 

[galactus]

Per the problem statement, the Earth is at a focus. It also says the center of the Earth is at the origin(0,0). Therefore, a focus is at the origin and will have coordinates (0,0).

The equation for an ellipse with major axis parallel to the x-axis and center (h,k) is:

\(\displaystyle \frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\)

If I am picturing the problem correctly, it appears the center of the ellipse will be at (150,0).

 

[soroban]

Hello, discreditedvalidity1

You did good . . . a few minor errors.

A satellite is in an elliptical orbit around the earth with the center of the earth at one focus.
The height of the satellite over the earth varies between 140 and 440 miles.
Assume the earth is a sphere with radius 3960 miles.
Find an equation for the path of the satellite with the origin at the center of the earth.


Okay, so far I have done this:
\(\displaystyle 2a\:=\:140\,+\,2(3960)\,+\,144\:=\:8500\)
\(\displaystyle \;a\,=\,4250\)

\(\displaystyle a\,-\,c\:=\:3960\,+\,140\:=\:4100\)
\(\displaystyle \;\;\;\,c\:=\:a\;-\,4100\:=\:4250\,-\,4100\:=\:150\)

Foci = \(\displaystyle (\pm150,0)\;\;\) . . . no

\(\displaystyle c^2\:=\:a^2\,-\,b^2\)
\(\displaystyle b^2\:=\:a^2\,-\,c^2\:=\:4250^2\,-\,150^2\:=\:18,040,000\)

That's all I have.
1) Have I done right so far? If not, what am I doing wrong?
2) How on earth do I get the equation?

   perigee                F     C                         apogee
      * - - - - - - - - - + - - o - - - - - - - - - - - - - +
   -4100                  |    150                       4400

The Earth is at F (a focus).
As Galactus pointed out, the center C is at \(\displaystyle \,(150,0)\)

You know: \(\displaystyle \,a^2\,=\,4250^2\,=\,18,062,500\,\) and \(\displaystyle \,b^2\,=\,18,040,000\)


Therefore: the equation is: \(\displaystyle \:\frac{(x\,-\,150)^2}{18,062,500}\,+\,\frac{y^2}{18,040,000} \;= \;1\)