Let's say we have an elliposoid given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$. The volume is given by $V = \frac{4}{3} \pi abc$.
If at a certain moment in time $a = 2, b = 3$, and $c= 4$, and $a$ is changing at a rate of $+2$ and $b$ is changing at a rate of $-3$, how do we go about finding the rate that $c$ must be changing by for the volume to be constant?
I think we have to take the partial derivative of $V$ with respect to $c$, but overall I am unsure how to approach the problem and help would be greatly appreciated.
hint: $V'(t) = 0 \iff V\text{ is constant} \Rightarrow a'bc + ab'c + abc' = 0$, and you are given : $a = 2, b = 3, c = 4, a' = 2, b' = -3$. Can you solve for $c'$?