Arocket, launched from the earth, shuts off its engines when its speed is 4.90 km/s.
it reaches a maximum altitude of 2570 km before it falls back to earth.

at what altitude did the rocket's engines shut off?

the mass and radius of the earth are 5.97 1024 kg and 6.37 106 m, respectively.

1346 km

Explanation:

If we analyze the motion on the Vertical direction we can calculate the distance traveled after the rocket shuts off its engine:

The acceleration is gravity, so:

so:

We took the acceleration of gravity as negative because it is going downward.

So the altitude when the rocket's engines shut off is:

it is a negative acceleration