√g/4 = √g[r/r+h]
1/2 = 6400/6400+h
6400+h=12800
h=6400km

Introduction:
The acceleration due to gravity on the surface of the Earth is approximately 9.8 m/s^2. This value decreases as we move away from the surface of the Earth due to the inverse square law. To find the distance at which the acceleration due to gravity becomes 25% of its value on the surface of the Earth, we can use the formula for gravitational acceleration and solve for the distance.

Formula:
The formula for gravitational acceleration is given by:

g = (G * M) / r^2

Where:
g is the acceleration due to gravity,
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
M is the mass of the Earth, and
r is the distance from the center of the Earth.

Step 1: Find the gravitational acceleration at the surface of the Earth:
We know that the acceleration due to gravity on the surface of the Earth is approximately 9.8 m/s^2. Plugging this value into the formula, we can solve for M/r^2.

9.8 = (G * M) / (6400 km)^2

Step 2: Find the distance at which the acceleration due to gravity is 25% of its value on the surface:
We need to find the distance r at which the acceleration due to gravity is 25% of its value on the surface. Let's call this value g'.

g' = 0.25 * 9.8

Now we can plug in this value into the formula and solve for r.

0.25 * 9.8 = (G * M) / r^2

Step 3: Solve for r:
We can rearrange the formula to solve for r:

r^2 = (G * M) / (0.25 * 9.8)

r^2 = 4 * (G * M) / 9.8

r = √[(4 * (G * M)) / 9.8]

Step 4: Calculate the distance:
Now we can calculate the distance by plugging in the values for G and M.

r = √[(4 * (6.67430 × 10^-11 m^3 kg^-1 s^-2) * M) / 9.8]

r = √[(4 * (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (5.972 × 10^24 kg)) / 9.8]

Calculating this expression will give us the distance from the surface of the Earth at which the acceleration due to gravity becomes 25% of its value on the surface.