Explanation

(c) The value of acceleration due to gravity at height $h$ (when $h$ is not negligible as compared to $R$)
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ g^{'} = g\frac{R^{2}}{{(R + h)}^{2}}$
Here, $\ g^{'} = \frac{g}{2}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \frac{g}{2} = g\frac{R^{2}}{(R + h)^{2}}$
$or\ \ \ \ \ \ \ \ \ \ \frac{1}{2} = \frac{R^{2}}{(R + h)^{2}}$
$or\ \ \ \ \ \ \ \sqrt{\frac{1}{2}} = \frac{R}{R + h}$
$or\ \ R + h = \sqrt{2}\ R$
$\therefore\ \ \ \ \ \ \ \ \ \ \ h = \left( \sqrt{2} - 1 \right)R$