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UnkleRhaukus

  • 3 years ago

How to optimise a cone ?

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  1. UnkleRhaukus
    • 3 years ago
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    minimise surface area to volume ratio

  2. UnkleRhaukus
    • 3 years ago
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    \[V_{\text{cone}}=\frac {\pi r^2}3\]

  3. UnkleRhaukus
    • 3 years ago
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    \[SA_\text{cone}=\]

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

  5. Jonask
    • 3 years ago
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    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

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