Problema de cálculo no puedo resolverlo

;)
Hola Anónimo!

$$\begin{align}&Sea\ p\ €/cm^2\ precio\ base\\&\\&Coste:\\&C=p2 \pi r h+3p \pi r^2\\&24 \pi= \pi r^2 h ==> h= \frac{24}{r^2}\\&\\&C(r)=p2 \pi r \frac{24}{r^2}+3p \pi r^2=p \pi ( \frac {48}{r}+3r^2)\\&\\&derivando\\&C'(r)=p \pi (- \frac {48}{r^2}+6r)\\&\\&C'(r)=0\\&==> - \frac {48}{r^2}+6r=0==>  \frac {48}{r^2}=6r==> r^3= \frac{48}6=8\\&\\&r=\sqrt[3] 8=2 \ cm\\&h= \frac{24}{2^2}=6\ \ cm\\&comprobacion \ mínimo:\\&\\&C''(r)=p \pi\Big(-48r^{-3}(-2)+6 \Big) =p \pi (- \frac{96}{r^3}+6)\\&C''(2)=p \pi(\frac {96}{2^3}+6) \ >0 ==> mínimo\end{align}$$