Show that the right circular cone of least curved surface and given volume has an altitude equal to\(\sqrt{2}\) time the radius of the base.
Let r and h be the radius and the height(altitude)of the cone respectively.
Then,the volume(V)of the cone is given as:
\(V=\frac{1}{3\pi}\)\(\pi r^{2}h⇒h=\frac{3V}{r^{2}}\)
The surface area\((S)\)of the cone is given by,
\(S=\pi rl\) (where \(l\) is the slant height)
\(=\pi r\sqrt{r^{2}+h^{2}}\)
\(=\pi r\sqrt{r^{2}+\frac{9V^{2}\pi}{/π^{2}r^{4}}}\)\(=\frac{r\sqrt{9^{2}r^{6}+V^{2}}}{\pi r^{2}}\)
\(=\frac{1}{r}\sqrt{π^{2}r^{6}+9V^{2}}\)
\(∴\frac{dS}{dr}\)\(=\frac{r.\frac{6\pi ^{2}r^{5}}{2\pi^{2}\sqrt{r^{6}9V^{2}}}-\sqrt{\pi ^{2}r^{6}+9V^{2}}}{r^{2}}\)
\(=\frac{3\pi^{2}r^{6}-\pi ^{2}r^{6}-9V^{2}}{r^{2}\sqrt{\pi 2r^{6}+9V^{2}}}\)
\(=\frac{2\pi^{2}r^{6}-9V^{2}}{r^{2}\sqrt{\pi^{2}r^{6}+9V^{2}}}\)
\(=\frac{2\pi^{2}r^{6}-9V^{2}}{r^{2}\sqrt{\pi^{2}r^{6}+9V^{2}}}\)
Now,\(\frac{dS}{dr}\)\(=0⇒2\pi^{2}r^{6}=9V^{2}⇒r^{6}=\frac{9V^{2}}{2\pi^{2}}\)
Thus,it can be easily verified that when \(r^{6}=\frac{9V^{2}}{2\pi^{2}},\frac{d^{2}S}{dr^{2}}>0.\)
By second derivative test, the surface area of the cone is the least when \(r^{6}=\frac{9V^{2}}{2\pi^{2}}\)
When \(r^{6}=\frac{9V^{2}}{2\pi^{2}}\)\(,h=\frac{3V}{\pi r^{2}}\)=\(\frac{3}{\pi r^{2}}(\frac{2\pi ^{2}r^{6}}{9})^{\frac{1}{2}}\)\(=\frac{3}{\pi r^{2}}.\frac{\sqrt{2}\pi r^{3}}{3}=\sqrt{2}r.\)
Hence,for a given volume,the right circular cone of the least curved surface has an
altitude equal to\(\sqrt{2}\) times the radius of the base.
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