The approximate change in the volume of a cube of side $x$ metres caused by increasing the side by $3\%$ is

(A) $0.06{\rm{ }}{x^3}{{\rm{m}}^3}$

(B) $0.6{x^3}{{\rm{m}}^3}$

(C) $0.09{x^3}{{\rm{m}}^3}$

(D) $0.9{x^3}{{\rm{m}}^3}$

Option C is correct

We know that the volume $V$ of a cube with edge $x$ is given by
$V = {x^3}$
$\Rightarrow$ $\cfrac{{dV}}{{dx}} = 3{x^2}$

Hence, $\Delta V \approx 3{x^2}\Delta x = 3{x^2}\left( {\cfrac{3}{{100}}x} \right)$
$= \cfrac{{9{x^3}}}{{100}}$

Therefore approximate change in volume is given by

$= \cfrac{{9{x^3}}}{{100}}{{\rm{m}}^3} = 0.09{x^3}{\rm{ }}{{\rm{m}}^3}$