Question

How high above the earth’s surface would you have to be before your weight would be only 0.300 its current value? (Note: The earth’s radius is 6378 km.)

Answer 1

Your weight is a force caused by gravity. Assuming your force on the Earth's surface (F1 @ r1) and your force above the earth's surface (F2 @ r2) you set:

F2 = 0.3 * F1
or
F2 / F1 = 0.3

Formula for gravitational force:

F1 = G * m1 * m2 / (r1)^2
F2 = G * m1 * m2 / (r2)^2

G, m1 and m2 (your and Earth's) masses are constant, so G, m1, and m2 will all cancel. Solving for F2/F1:

F2 / F1 = (r1)^2 / (r2^2) = 0.3

Now solve for r2:

r1 / (0.3)^1/2 = r2 = 11644

That is your radius from the earth center, so you would be 11644-6378 = 5267 km above the surface​