Question

Assume there is a cube with all sides equal to x. The change in x with respect to t is x(t)=2t. What is the formula for the rate of change of the volume of the cube?

Answer 1

$\bf \textit{volume of a cube}\\\\ V=x^3~~ \begin{cases} x=length~of\\ \qquad a~side\\[-0.5em] \hrulefill\\ x(t)=2t\\ x=2t \end{cases}\implies V=[x(t)]^3\implies V=[2t]^3 \\\\\\ V=[2^3t^3]\implies V=8t^3\implies \boxed{\cfrac{dV}{dt}=24t^2}$