calculus optimization for the volume of a cone A circular piece of card with a sector removed is folded to form a conte.  The slanted height of the cone is 12cm and the vertical height is h.
Show that the volume of the cone ${Vcm^2}$ is given by the expression 
${V = {1\over 3}\pi h(144 - h^2)}$
The volume of a cone is ${{1\over 3}\pi r^2h}$
${3 = \pi r^2h}$
${h = {3\over {\pi r^2}}}$
I would then plug h into the original volume equation:
${{1\over 3}\pi r^2{3\over {\pi r^2}}}$
Which is obviously far removed from the original question.
I don't get where the equation in the question comes from
 A: Hint:
the slanted height ($12$) is the hypotenuse of a rectangular triangle that has, as other sides, the height ($h$) and the radius ($r$) of the basis of the cone. So $144-h^2=r^2$
A: The volume of a cone can be calculated with $h$ and the area $A$ of the base circle. Comparing this to the given formula, you can see what part corresponds to $A$:
$$
\begin{array}{rlc}
V&=\frac13 h&A\\
V &= {1\over 3}h&\overbrace{\pi (144 - h^2)}
\end{array}
$$
The formula to calculate the area of a circle involves the radius $r$ of the circle. The same principle applies again: find the parts of the formula that correspond to those in the general formula:
$$
\begin{array}{rlc}
A&=\pi &r^2\\
A &= \pi &\overbrace{(144 - h^2)}
\end{array}
$$
There's $r^2$ and $h^2$ which strongly hint towards an old Greek man's famous formula. What's the third term involved? $144$ which looks like... yup, there you go: $144 = 12^2$
One last time, the general formula for the Pythagorean Theorem that involves a minus or in other words, calculates one of the shorter sides from the hypothenuse and the other shorter side looks like this. Again, looking for the parts in the formula that you have that correspond to those in the general formula:
$$
\begin{array}{rcl}
a^2=&c^2 &-b^2\\
r^2 =& \overbrace{\underbrace{144}_{12^2}} &- h^2
\end{array}
$$
As you can see, the given formula stems from the right sided triangle that's formed by the slanted height, height and base circle radius of the cone.
