At what height above the surface of the Earth does the force of gravity decrease by 10%?

At what height above the surface of the Earth does the force of gravity decrease by 10%? The radius of the Earth is considered equal to 6400 km.

Initial data: F1 (gravity at height h) = 0.9F, where F is the gravitational force on the Earth’s surface.

Constants: R (Earth radius) = 6400 km (6400000 m).

Gravity on the Earth’s surface:

F = G * m * M / R ^ 2.

Gravity at height h from the Earth’s surface:

F1 = G * m * M / (R + h) ^ 2.

Ratio:

0.9 * G * m * M / R ^ 2 = G * m * M / (R + h) ^ 2

0.9 / R ^ 2 = 1 / (R + h) ^ 2

(R + h) ^ 2 = R ^ 2 / 0.9.

R + h = √ (R ^ 2 / 0.9).

h = √ (0.9 * R ^ 2) – R = √ (6400000 ^ 2 / 0.9) – 6400000 = 346192 m ≈ 346 km.