Answer
$\dfrac{dv}{dt}=\dfrac{1}{6}$
Work Step by Step
$\dfrac{dr}{dt}+2s\dfrac{ds}{dt}+3v^2\dfrac{dv}{dt}=0$
As we are given that ${r+s^2+v^3=12}$,$\dfrac{dr}{dt}=4$ and $\dfrac{ds}{dt}=-3$
Thus, $4+2s(-3)+3v^2\dfrac{dv}{dt}=0$
Plug $r=3$ and $s=1$
Then ${r+s^2+v^3=12}=3+1+v^3$
or, $v=2$
or, $4+2\times1(-3)+12 \dfrac{dv}{dt}=0$
and $\dfrac{dv}{dt}=\dfrac{1}{6}$
