The problem states that F=k/r2 where F is the gravitational force (same as the weight of the astronaut), r is the distance between the object and the center of the earth, and k is the constant of proportionality. We need to find k first to solve the problem.
We know the astronaut weights 209 pounds wen he/she is 4000 miles from the center of the earth (standing on the earth's surface).
Solving for k we have
To find the weight of the astronaut at 4300 miles above the earth (or 4000+4300=8300 miles from the center of the earth) we plug this distance into the equation along with the determined value for k.
So the astronaut weighs 48.5 pounds when 4300 miles above the earth's surface.
Note: astronauts in orbit "feel" weightless because this gravitational force is offset by the centripetal force of the orbit (like the outward force on a merry-go-round)