Answer: 6,400 km
Explanation:
The weight of a person is given by:
[tex]W=mg[/tex]
where m is the mass of the person and g is the acceleration due to gravity. While the mass does not depend on the height above the surface, the value of g does, following the formula:
[tex]g=\frac{GM}{r^2}[/tex]
where
G is the gravitational constant
M is the Earth's mass
r is the distance of the person from the Earth's center
The problem says that the person weighs 800 N at the Earth's surface, so when r=R (Earth's radius):
[tex]800 N= W=mg=m \frac{GM}{R^2}[/tex] (1)
Now we want to find the height h above the surface at which the weight of the man is 200 N:
[tex]200 N = W' = mg' = m \frac{GM}{(R+h)^2}[/tex] (2)
If we divide eq.(1) by eq.(2), we get
[tex]\frac{800 N}{200 N}=\frac{W}{W'}=\frac{(R+h)^2}{R^2}[/tex]
[tex]4=\frac{(R+h)^2}{R^2}[/tex]
By solving the equation, we find:
[tex]4R^2 = (R+h)^2=R^2+2Rh+h^2\\h^2 +2Rh-3R^2 =0[/tex]
which has two solutions:
[tex]h=-3R[/tex] --> negative solution, we can ignore it
[tex]h=R[/tex] --> this is our solution
Since the Earth's radius is [tex]R=6.4\cdot 10^6 m[/tex], the person should be at [tex]h=R=6.4\cdot 10^6 m=6400 km[/tex] above Earth's surface.
