Answer
The altitude is $~~420~km$
Work Step by Step
We can find the original energy of the system:
$E = -\frac{GMm}{2r}$
$E = -\frac{(6.67\times 10^{-11}~N~m^2/kg^2)(5.98\times 10^{24}~kg)(220~kg)}{(2)(6.37\times 10^6~m+6.40\times 10^5~m)}$
$E = -6.25\times 10^9~J$
We can find the energy that is lost in 1500 revolutions:
$(1500)(1.4\times 10^5~J) = 2.1\times 10^8~J$
We can find the energy in the system after 1500 revolutions:
$E = -6.25\times 10^9~J - 2.1\times 10^8~J = -6.46\times 10^9~J$
We can find the orbital radius after 1500 revolutions:
$E = -\frac{GMm}{2r}$
$r = -\frac{GMm}{2E}$
$r = -\frac{(6.67\times 10^{-11}~N~m^2/kg^2)(5.98\times 10^{24}~kg)(220~kg)}{(2)(-6.46\times 10^9~J)}$
$r = 6.79\times 10^6~m$
We can find the altitude $h$ above the Earth's surface:
$h = 6.79\times 10^6~m-6.37\times 10^6~m$
$h = 4.2\times 10^5~m$
$h = 420~km$
The altitude is $~~420~km$
