Given,
Distance from the surface to the center of the earth, d=4000 miles
Distance from the center to you at a height of 8000 miles, a= 8000+4000=12000 miles
The gravitational force acting on a person at the surface is equal to his weight.
From Newton's Universal Law of Gravitation, the gravitational force is
![F=\frac{G\times M\times m}{r^2}](https://tex.z-dn.net/?f=F%3D%5Cfrac%7BG%5Ctimes%20M%5Ctimes%20m%7D%7Br%5E2%7D)
Where G is the gravitational constant, M is the mass of the earth, m is the mass of the object/person, r is the distance between the center of the earth and the object/person
At the surface, this force is equal to the weight of the person, W=mg
i.e.
![F_s=\frac{G\times M\times m}{d^2}=W](https://tex.z-dn.net/?f=F_s%3D%5Cfrac%7BG%5Ctimes%20M%5Ctimes%20m%7D%7Bd%5E2%7D%3DW)
On substituting the of d,
![W=\frac{\text{GMm}}{4000^2}](https://tex.z-dn.net/?f=W%3D%5Cfrac%7B%5Ctext%7BGMm%7D%7D%7B4000%5E2%7D)
At a height of 8000 miles from the surface, the gravitational force is equal to,
![F_a=\frac{GMm}{12000^2}](https://tex.z-dn.net/?f=F_a%3D%5Cfrac%7BGMm%7D%7B12000%5E2%7D)
On dividing the above two equations,
![\frac{F_a}{W}=\frac{4000^2^{}}{12000^2}=\frac{1}{9}](https://tex.z-dn.net/?f=%5Cfrac%7BF_a%7D%7BW%7D%3D%5Cfrac%7B4000%5E2%5E%7B%7D%7D%7B12000%5E2%7D%3D%5Cfrac%7B1%7D%7B9%7D)
Therefore,
![F_a=\frac{1}{9}W](https://tex.z-dn.net/?f=F_a%3D%5Cfrac%7B1%7D%7B9%7DW)
Therefore at a height of 8000 miles above the surface of the earth, the force of gravity becomes 1/9 time your weight.