Question

The rate of change of volume of a cube when its body diagonal is $6\sqrt{3}$ is equal to

• A. 108
• B. 6
• C. 216
• D. 12

Answer 1

The correct option is A 108

Let the side of the cube be a

Volume = $a^3$

Rate of change of volume with respect to the side = $\frac{d\left(a^3\right)}{da}$
$=3a^2$

Body diagonal of the cube of side a = $a\sqrt{3}$

Given, the body diagonal of the cube = $6\sqrt{3}$
$\Rightarrow a\sqrt{3}=6\sqrt{3}$
$\Rightarrow a=6$

Thus, the rate of change of volume = $3\times 6^2$ = 108.