A satellite in an elliptical orbit has a speed of 9.00km/s when it is at its closes approach to the Earth(perigee). The satellite is 7.00x10^6 m from the center of the Earth at this time. When the satellite is at its greatest distance from the center of the Earth (apogee), its speed is 3.66km/s. Find the distance from the satellite to the center of the Earth at apogee. (assume any energy losses are negligible.)

Post your work thus far and I'll continue/correct/guide

Kfinal + Ugravfinal = Kinitial + Ugravinitial

which i ended up with
Vfinal^2 + [2GM/r] = Vinitial^2 + [2GM/r]

I believe I need to solve for r but this is where I got stuck.

Well since it's an elliptical orbit you can set them the distances as functions of Kvff^3

<Kf^3>(2GM^4)=vf%
vf%<6.76[FL(3.33/Ki)]=d%
d%=<4.1882[Ug]
d=(1.333<pi^4>)(d%)

Tell me what you get

Quick explanation because I have to go;

Kvff^3
Rearrange values so that kf^3=vf%
vf% is of 6.76 because of orbit while 3.33 is the continual. Solve for d%
d% the continual converted to elliptical motion is going to be roughly 4.1882 (4.188? I believe it's two, but shouldn't matter)
By Ugravinitial
and you have d when you apply the standard orbital motion functions.

To find the distance from the satellite to the center of the Earth at apogee, we can start by using the conservation of mechanical energy.

The conservation of mechanical energy states that the sum of the kinetic energy and the potential energy of an object remains constant as long as no external forces act on the object.

At perigee, the satellite's speed is 9.00 km/s, which means its kinetic energy is given by:

Kinetic energy at perigee = (1/2) * mass * velocity^2

We are not given the mass of the satellite, but since we only need the ratio of the distances, the mass cancels out, so we can ignore it for now.

The potential energy at perigee is given by:

Potential energy at perigee = - G * mass * M / distance at perigee

where G is the gravitational constant and M is the mass of the Earth.

Therefore, the total mechanical energy at perigee is:

Total energy at perigee = Kinetic energy at perigee + Potential energy at perigee

Now we can calculate the mechanical energy at perigee using the given values.

Next, we can calculate the mechanical energy at apogee. The speed at apogee is given as 3.66 km/s, and the potential energy at apogee is given by:

Potential energy at apogee = - G * mass * M / distance at apogee

Since the mechanical energy is conserved, we can equate the total energy at perigee to the total energy at apogee:

Total energy at perigee = Total energy at apogee

Solving this equation will give us the distance from the satellite to the center of the Earth at apogee.