If the 78.0 kg astronaut were in a spacecraft 6R from the center of the earth, what would the astronaut's weight be on earth?

His weight on Earth doesn't depend on where he is at the moment.

Weight = (mass) x (gravity)

On the Earth's surface, the acceleration of gravity is 9.8 m/s².

So, on the surface, the astronaut weighs

(78 kg) x (9.8 m/s²) = 764.4 Newtons (about 172 pounds).

No matter where he goes, his mass doesn't change.

But his weight does.

If he's 6R from the enter of the Earth, then he's 5 times as far from it

as when he's standing on the surface. The acceleration of gravity

out there is

(1 / 5²) = 1/25 = 0.04 times

the acceleration of gravity on the surface.

His weight out there is

(78 kg) x (9.8/25 m/s²) = 30.6 Newtons (about 6.9 pounds)

Acceleration due to gravity on the surface of radius R = GM/R^2 = 9.8 m/s^2

weight at surface of earth = mg = 78 x 9.8 = 764.4 N

acceleration due to gravity at 6R = GM / (6R) ^2

= GM/R^2 / 36

=9.8/36 = 0.27 m/s^2

weight on spacecraft = 78 x 0.27 = 21.2 N