Answer:
Approximately [tex]384.2\; {\rm km}[/tex].
Explanation:
The distance travelled during each orbit, [tex]s[/tex], can be found from the orbital speed and the period of the orbit. Note the unit conversion from kilometers per hour to kilometers per second:
[tex]\displaystyle v = 27600\; {\rm km\cdot h^{-1}} \times \frac{1\; {\rm h}}{3600\; {\rm s}} = \frac{23}{3}\; {\rm km\cdot s^{-1}}[/tex].
[tex]\displaystyle s = v\, t = \left(\frac{23}{3}\; {\rm km\cdot s^{-1}}\right)\, \left(5560\; {\rm s}\right) = \frac{127\, 880}{3}\; {\rm km}[/tex].
Assuming that the orbit is circular, divide the distance travelled during each orbit (the circumference) by [tex]2\, \pi[/tex] to find the radius of the orbit:
[tex]\begin{aligned}r &= \frac{(\text{circumference})}{2\, \pi} \\ &= \frac{\displaystyle \frac{127\, 880}{3}\; {\rm km}}{2\, \pi} \\ &= \frac{63940}{3\, \pi}\; {\rm km}\end{aligned}[/tex].
Note that this radius gives the distance between the space station and the center of the planet. To find the distance between the space station and the surface of the planet (altitude), subtract the radius of the planet from the radius of the orbit:
[tex]\displaystyle \frac{63\, 940}{3\, \pi}\; {\rm km} - 6400\; {\rm km} \approx 384.2\; {\rm km}[/tex].
(Rounded to one decimal place.)
In other words, the distance between the space station and the surface of the planet would be approximately [tex]384.2\; {\rm km}[/tex].
