The approximate height from the surface of earth at which the...

The variation of the acceleration due to gravity with height $h$ is given by,

${g}^{'} = \dfrac{g}{{\left(1 + \dfrac{h}{R}\right)}^{2}}$.

Since the weight at given height is $\dfrac{1}{3}$ of the weight at the surface

$⇒ {g}^{'} = \dfrac{g}{3} ⇒ \dfrac{g}{{\left(1 + \dfrac{h}{R}\right)}^{2}} = \dfrac{g}{3} ⇒ \left(1 + \dfrac{h}{R}\right) = \sqrt{3} ⇒ h = R\left(\sqrt{3} - 1\right) = 6400 \times \left(1 . 732 - 1\right) = 4685km$