On the way to the moon, the Apollo astronauts reach a point where the Moon’s gravitational pull is stronger than that of Earth’s. Find the distance of this point from the
center of the Earth. The masses of the
Earth and the Moon are 5.98 × 1024 kg and
7.36 × 1022 kg, respectively, and the distance
from the Earth to the Moon is 3.84 × 108 m.

3.456 × 10^(8) m

Explanation:

We are given;

Mass of earth; m_e = 5.98 × 10^(24) kg

Mass of moon; m_m = 7.36 × 10^(22) kg

Distance from earth to moon; L = 3.84 × 10^(8) m.

We are told that the Moon’s gravitational pull is stronger than that of Earth’s.

Thus;

F_EA ≤ F_MA

Formula for force due to gravity is;

F = GMm/r²

Applying to this question, we have;

(Gm_e•m_a)/x² ≤ (Gm_m•m_a)/(l - x)²

Where x is his distance from the center of the Earth

Now, G and m_a will cancel out from both sides and we plug on other values to get;

(5.98 × 10^(24))/x² ≤ (7.36 × 10^(22))/(3.84 × 10^(8) - x)²

Taking square root of both sides gives;

2445403852127.4966/x ≤ 271293199325.0107/(3.84 × 10^(8) - x)

Rearranging gives;

2445403852127.4966(3.84 × 10^(8) - x) ≤ 271293199325.0107x

Simplifying this gives;

9.0139(3.84 × 10^(8) - x) ≤ x

(34.613 × 10^(8)) - 9.0139x ≤ x

(34.613 × 10^(8)) ≤ x + 9.0139x

(34.613 × 10^(8)) ≤ 10.0139x

x ≥ (34.613 × 10^(8))/10.0139

x ≥ 3.456 × 10^(8)