Imagine an astronaut with a mass of 70 kg floating in space 10.0 m away from the center of gravity of an apollo command module whose mass is 6.00× kg determine the gravitational force acting on, and the resulting acceleration ( at that instant) of both the ship and person

The gravitational force between 2 objects is  G (M₁ x M₂) / R²

G = 6.67 x 10⁻¹¹ Nt-m² / kg²
M₁ = 70 kg 
M₂ = 6 x 10³ kg
R = 10

Force = (6.67 x 10⁻¹¹) (70) (6 x 10³) / (10)² = 2.8014 x 10⁻⁷ newton

The forces on the astronaut and on the module are equal.
That's the 'weight' of each one, in the presence of the other one.
(It's almost exactly 1 micro-ounce.)

Acceleration:  F = M A    ===>    A = F / M

Of the astronaut:

A = (2.8014 x 10⁻⁷) / 70 = 0.000 000 004 002 m/s²  (4.002 x 10⁻⁹ m/s²)

Of the command module:

A = (2.8014 x 10⁻⁷) / 6,000 = 0.000 000 000 04 669 m/s²  (4.669 x 10⁻¹¹ m/s²)