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Problem 17 Medium Difficulty

The surface area of a right-circular cone of radius $r$ and height $h$ is $S=\pi r \sqrt{r^{2}+h^{2}}$ , and its volume is $V=\frac{1}{3} \pi r^{2} h_{\text { . }}$
\begin{equation}\begin{array}{l}{\text { (a) Determine the ratio } h / r \text { for the cone with given surface area } S \text { and }} \\ {\text { maximum volume } V .} \\ {\text { (b) What is the ratio } h / r \text { for a cone with given volume } V \text { and mimimum }} \\ {\text { surface area } S ?} \\ {\text { (c) Does a cone with given volume } V \text { and maximum surface area exist? }}\end{array}\end{equation}

Answer

a) $\sqrt{2}$
b) $\sqrt{2}$
c) Therefore, the maximum surfaces area does not exist to the cone with given volume $V$ .

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Video Transcript

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