Question
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) $0.9{x^3}{{\rm{m}}^3}$
Application of Derivatives — Class 12 Maths Solution
Step-by-step Solution
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
$= \cfrac{{9{x^3}}}{{100}}{{\rm{m}}^3} = 0.09{x^3}{\rm{ }}{{\rm{m}}^3}$
NCERT & Exemplar solution for CBSE Class 12 Mathematics, Application of Derivatives. Curated by Sachin Sharma. Free for all students.