(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A right circular cone of base radius r and height h has a total surface area S and volume V . Show that 9V^{2}=r^{2}(S^{2}-2pi^{r2}S) . (i can do this part) . Hence or otherwise , show that for a fixed surface area S , the maximum volume of the cone occurs when its semi-vertical angle , theta is given by tan theta=1/2(root 2)

3. The attempt at a solution

From the proven equation ,

9V^{2}=r^{2}(S^{2}-2pi^{r2}S)

Differentiate this wrt to r ,

dV/dr=(2S^{2}r-8pi Sr^{3})/(18V)

dV/dr=0 , S=4pi r^{2}, substitue S with the area of cone , then

tan theta=r/h=1/(2 root 2)

This is my question , how do i prove that its a maximum ?

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# Homework Help: Optimization problem

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