OK, let's learn some physics and do this the easy way shall we?

Invoking that g ~ 1/r^2 we can write g'/g = (r/R)^2 = (1/16 g/g) = 1/16. So R = r sqrt(16) = 4r In which case h = 4r - r = 3r = 3*6400 = 19200 km above the surface. ANS.

g ~ 1/r^2 means the gravity field strength is inversely proportional to the square of the distance from the gravity source, r = 6400 km is the Earth's radius .

Gravitational Force (weight in this case):

F = G*m*M_earth / (R_earth + h)^2

Solve for m

m = F*(R_earth+h)^2/ (G*M_earth)

The mass doesn't change, it is constant

m = F*(R_earth+h)^2/ (G*M_earth) = constant

Being constant means (2) points are equal. What is changing are the force and the distance from the surface, h.

F1*(R_earth + h1)^2 / (G*M_earth) = F2*(R_earth + h2)^2 / (G*M_earth)

The G*M_earth's cancel

F1*(R_earth + h1)^2 = F2*(R_earth + h2)^2

On the initial side, you start at the surface, h1 = 0

F1*R_earth^2 = F2*(R_earth + h2)^2

On the final side, your weight is 1/16 the starting weight, F2 = 1/16*F1

F1*R_earth^2 = 1/16*F1*(R_earth+h2)^2

16*R_earth^2 = (R_earth+h2)^2

Expand the brackets

16*R_earth^2 = R_earth^2 + 2*R_earth*h2 + h2^2

h2^2 + 2*R_earth*h2 - 15*R_earth^2 = 0

Plug in R_earth = 6400 km

h2^2 + 12800*h2 - 614400000 = 0

Solve via the quadratic equation

http://www.math.com/students/calculators/source/qu...

h2 = 23400 km

That's what I got. Double check to make sure the numbers are correct. Hope my logic helped you understand how to do the problem.

EARTH

g = G x M(earth) / d^2

where g = Acceleration of gravity

G = universal gravitatonal constant 6.67259e-11

M = mass earth 5.98e24 kg

d = distance to center 6.38e6 meters

LOCATION DISTANCE g VALUE

Earth's surface 6.38 x 106 m 9.8

1000 km above surface 7.38 x 106 m 7.33

2000 km above surface 8.38 x 106 m 5.68

3000 km above surface 9.38 x 106 m 4.53

4000 km above surface 1.04 x 107 m 3.70

5000 km above surface 1.14 x 107 m 3.08

6000 km above surface 1.24 x 107 m 2.60

7000 km above surface 1.34 x 107 m 2.23

8000 km above surface 1.44 x 107 m 1.93

9000 km above surface 1.54 x 107 m 1.69

10000 km above surface 1.64 x 107 m 1.49

50000 km above surface 8.35 x 107 m 0.13