I have some past exams that I'm practicing before the real thing. They didn't come with answers and I'm really stumped on this one question:
"The volume V of a cylinder of radius r and height h is given by $V=\pi r^2 h$. Using differentials, find an approximate formula for the percentage increase in V in terms of the percentage increase in r and h. That is, find an approximate formula for $\frac {δV}{V}$ in terms of $\frac {δr}{r}$ and $\frac {δh}{h}$."
Any help would be greatly appreciated!
For fixed $h$ we have:
$\delta V=\dfrac{\partial V}{\partial r} \delta r=2\pi hr$ , so $\dfrac{\delta V}{V}=\dfrac{2\pi h r \delta r}{\pi r^2 h}=2\dfrac{\delta r}{r} $.
For fixed $r$:
$\delta V=\dfrac{\partial V}{\partial h} \delta h$ and: $\dfrac{\delta V}{V}=\dfrac{\pi r^2 \delta h}{\pi r^2 h}=\dfrac{\delta h}{h} $.