as you go above earths surface, the acceleration due to gravity will decrease. find the height, i meters, above the earths surface where this value will be 1/410g

1.23*10⁸ m

Explanation:

Acceleration due to gravity is:

a = GM / r²

where G is the universal gravitational constant,

M is the mass of the planet,

and r is the distance from the center of the planet to the object.

When the object is on the surface of the Earth, a = g and r = R.

g = GM / R²

When the object is at height i above the surface, a = 1/410 g and r = i + R.

1/410 g = GM / (i + R) ²

Divide the first equation by the second:

g / (1/410 g) = (GM / R²) / (GM / (i + R) ²)

410 = (i + R) ² / R²

410 R² = (i + R) ²

410 R² = i² + 2iR + R²

0 = i² + 2iR - 409R²

Solve with quadratic formula:

i = [ - 2R ± √ ((2R) ² - 4 (1) (-409R²)) ] / 2 (1)

i = [ - 2R ± √ (1640R²) ] / 2

i = (-2R ± 2R√410) / 2

i = - R ± R√410

i = (-1 ± √410) R

Since i > 0:

i = (-1 + √410) R

R = 6.37*10⁶ m:

i ≈ 1.23*10⁸ m