Explanation

(b) The value of acceleration due to gravity at height $h$ above the surface of the earth is given by
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$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g^{''} = \frac{g}{\left( 1 + \frac{h}{R} \right)^{2}}$
$\ \ \ \ \ g^{''} = g\left( 1 + \frac{h}{R} \right)^{- 2} = g\left( 1 - \frac{2h}{R} \right)$
$Given,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g^{''} = \frac{g}{4}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{g}{4} = g\left( 1 - \frac{2h}{R} \right)$
$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{4} = 1 - \frac{2h}{R}$
$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{2h}{R} = \frac{3}{4}$
$\Longrightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h = \frac{3R}{8}$