Question

How to optimise a cone ?

Answer 1

minimise surface area to volume ratio

Answer 2

\[V_{\text{cone}}=\frac {\pi r^2}3\]

Answer 3

\[SA_\text{cone}=\]

Answer 4

\[V=\frac{ 1 }{ 3 }\pi r^2 h,h \rightarrow \frac{ 3V }{ \pi r^2}\]\[S=\pi r h_{slant}\]
\[S=\pi r \sqrt{r^2+h^2}\]
\[S=\pi r \sqrt{r^2+\frac{ 9V }{ \pi^2r^4 }}\]
\[S (r)=\sqrt{r^4\pi^2+\frac{ 9V }{ r^2 }}\]
\[S'(r)=\frac{(2r^3\pi^2-9Vr^{-3}) }{ \sqrt{r^4\pi^2+\frac{ 9V }{ r^2 }} }\]
\[S'(r)=\frac{(2r^3\pi^2-9Vr^{-3}) }{ S }\]

Answer 5

Hence nominator equal to 0
\[2r^3\pi^2=\frac{ 9V }{ r^3 }\]
\[r^6=\frac{ 9V }{2\pi^2 } \]
i am not sure if this is what you want,,,tried