Optimization problem, stuck

• Nov 9th 2006, 06:12 PM
aaasssaaa
Optimization problem, stuck
A conical drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup

link of picture of the cup is found in the attachement
• Nov 10th 2006, 03:52 AM
Soroban
Hello, aaasssaaa!

Quote:

A conical drinking cup is made from a circular piece of paper of radius $\displaystyle R$
by cutting out a sector and joining the edges $\displaystyle CA$ and $\displaystyle CB.$
Find the maximum capacity of such a cup.

Code:

            ..*.*.*..         .*:::::::::::*.       A*:::::::::::::::*B       *  *:::::::::::*  *           R *:::::::* R       *      *:::*      *       *        *        *       *        C        *       *                *         *              *           *          *               * * *

The side view of the cup looks like this:
Code:

              r       *-----+-----*       \    :    /         \  h:  /         \  :  / R           \ : /           \:/             *

where $\displaystyle r$ is the radius of the cone and $\displaystyle h$ is its height.

We see that: .$\displaystyle r^2 + h^2\;=\;R^2\quad\Rightarrow\quad r^2 \;=\;R^2 - h^2$ [1]

The volume of a cone is: .$\displaystyle V \;=\;\frac{\pi}{3}r^2h$ [2]

Substitute [1] into [2]: .$\displaystyle V \;=\;\frac{\pi}{3}\left(R^2-h^2\right)h\quad\Rightarrow\quad V \;=\;\frac{\pi}{3}\left(Rh - h^3\right)$

Maximize $\displaystyle V:\;\;V' \;=\;\frac{\pi}{3}\left(R - 3h^2\right)\;=\;0\quad\Rightarrow\quad h^2 \:=\:\frac{R^2}{3}$

. . and we get: .$\displaystyle \boxed{h \:= \:\frac{\sqrt{3}R}{3}}$

Substitute into [1]: .$\displaystyle r^2\;=\;R^2 - \left(\frac{R}{\sqrt{3}}\right)^2\quad\Rightarrow\ quad\boxed{ r^2\;=\;\frac{2R^2}{3}}$

Substitute into [2]: .$\displaystyle V \;=\;\frac{\pi}{3}r^2 h\;=\;\frac{\pi}{3}\left(\frac{2R^2}{3}\right)\lef t(\frac{\sqrt{3}R}{3}\right)$

Therefore, the maximum volume is: .$\displaystyle V \;=\;\boxed{\frac{2\pi\sqrt{3}}{27}R^3}$

• Nov 10th 2006, 03:09 PM
aaasssaaa
Hey what program did you use to right your numbers, roots and pie in
• Nov 11th 2006, 12:15 AM
CaptainBlack
Quote:

Originally Posted by aaasssaaa
Hey what program did you use to right your numbers, roots and pie in

The mathematical typesetting here is done using a LaTeX see this
(to see the code generating the equations quote the message and
you will see the typesetting codes embedded in the message).

RonL