At what point the certain object feels zero garvitational force between earth and the moon. take the required information from the book. required: distance away from earth or moon

Newton gives us force

F=-G*M*m/R^2

Divide by the mass of the object to get acceleration

a=F/m = GM/R^2

You are solving the distance where the force from the moon equals the force from the Earth -

a= G*(mass of earth)/R^2 - G*(mass of moon)/(384400000-R)^2=0

Or in other words

G*(mass of earth)/R^2=G*(mass of moon)/(384400000-R)^2

Divide by G*(mass of the moon) and multiply by R^2 to get terms on both sides

G*(mass of earth)/(G*(mass of moon))=R^2/(384400000-R)^2

Since the mass of the moon and earth are fixed, the left side is a number

81.3 = R^2/(384400000-R)^2 → R^2/(384400000-R)^2–81.3 = 0

simplify

R/(384400000-R) = 9.01665

R=346024000 = 346,024 km.

Explanation: