We will need to account for the speed necessary for the satellite to go into a circular orbit, so let's find that speed first.
We know that the only force to act on the satellite is the Earth's gravity, so the force of gravity is equal to the centripetal force on the satellite:
where m is the mass of the satellite and r is the distance from the center of the Earth to the satellite. (r is NOT 16000 km!)
Solving for v I get:
(You can calculate out what number this is, I'm going to leave it as a formula for now. As a reference, it's about using .)
Now, for the launch. We know what speed we want at the end, thus we know what final kinetic energy we want left over.
So I'm going to set an origin where the rocket takes off and let +y be upward. Gravity is a conservative force, so there are no non-conservative forces present. (I'm assuming the satellite is just thrust upward all at once instead of being in a rocket thrusting it upward over time.)
(where I have set the 0 point for gravitational potential energy to be at infinity.)
We know the final speed, v, and we want , (and note that , not 0.)
You can now plug your numbers in and get the answer. I'm going to stick that ugly expression in for v and see what comes out:
So I get:
which is more or less what your given answer is.