A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is a circumference.)
My answer:
$$ \begin{align*} 2\pi \cdot r +h&= 108\\ V(r,h) &= \pi r^2 \cdot h \quad \Rightarrow \quad V(r) = \pi r^2 \cdot (108 - 2 \pi r)\\ \\ V'(r) &= 0 \quad \Rightarrow \quad -6\pi r^2+2\pi r \cdot 108 = 0 \end{align*} $$
since $ r \neq 0 $ we have $r = \dfrac{36}{\pi}$ and $h = 36$.
Is my reasoning ok?
Thank you.
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