Question

Maximizing the volume of a cone with radius 12cm and determine the angle (A) needed to create this.

Answer 1

A cone, as drawn above, has a volume given by
\[V=\frac{ 1 }{ 3 }*\pi*r ^{2}*h\]
Since
\[\cot \left( \Theta \right)=\frac{ h }{ r }\]
we can rewrite the formula in the following steps:
\[V=\frac{ 1 }{ 3 }*\pi*r ^{2}*h*\frac{ r }{ r }\]
\[V=\frac{ 1 }{ 3 }*\pi*r ^{3}*\frac{ h }{ r }\]
\[V=\frac{ 1 }{ 3 }*\pi*r^3*\cot \left( \Theta \right)\]

Using this formula (and considering r as a constant 12cm): as theta gores toward zero (from some positive value), V gets larger and larger toward infinity (which would correspond to height going toward infinity. However, since theta can never actually have a value of zero due to physical impossibility, there is no maximum value for the volume.

Answer 2

Ah, you beat me to it manetbo! My browser refreshed and I lost my picture and everything haha! I will second your answer! Nice write up!!