Understanding the Concepts

* Gravitational Acceleration: The acceleration due to gravity (g) decreases with distance from the center of the Earth.

* Inverse Square Law: The force of gravity, and therefore the acceleration due to gravity, follows an inverse square law. This means that if you double the distance, the acceleration becomes four times weaker.

Setting up the Problem

Let:

* *R* be the radius of the Earth.

* *g* be the acceleration due to gravity at the Earth's surface.

* *r* be the distance from the center of the Earth to the point where the acceleration is 1/45 of its value at the surface.

Using the Inverse Square Law

We know that the acceleration due to gravity is inversely proportional to the square of the distance from the center of the Earth. So:

g/g' = (r')²/r²

Where:

* g' is the acceleration at the new distance (1/45 * g)

* r' is the new distance from the Earth's center

Solving for r'

1. Substitute the known values:

(g) / (1/45 * g) = (r')² / R²

2. Simplify:

45 = (r')² / R²

3. Solve for r':

r'² = 45R²

r' = √(45R²)

r' = √45 * R

4. Approximate the square root:

r' ≈ 6.7 * R

Therefore, the distance from the center of the Earth to the point where the acceleration due to gravity is 1/45 of its value at the surface is approximately 6.7 times the radius of the Earth.