Building a Goblet

jacksonpeeble

1. The problem statement, all variables and given/known data
As our final project, we have been instructed to design a goblet (which will actually be physically prototyped using a 3-D printer) according to the following specifications:
1. The goblet must hold exactly 200 cc.
2. The goblet must be symmetric (a solid of revolution).
3. The goblet requires exactly 150 cc of material to manufacture.
4. The stem thickness must be at least .25cm at its thinnest point.


2. Relevant equations
Disc Method


3. The attempt at a solution
I realize that this is very in depth. I would just appreciate step-by-step guidance as to what to do, and I will post my findings and we can all move on to the next step. This is my final project, and worth a lot, but I also want to produce a high-quality end product.

After sketching several models, I decided on one that uses a base followed by a sine graph:
f1(x)=.25sin(x) | -6<x<1.26966
followed by a logarithmic graph (I found that natural log was the most aesthetically-pleasing):
f2(x)=ln(x) | 1.26966<x<8
and a closing graph (for the hole in the middle):
f3(x)=ln(x) | 1.26966+1/6<x<8
where 1.26966 is an essentially arbitrary value that can be shifted (I merely picked it because it is the readily-available location of intersection that I graphed on my calculator). The graphs themselves are easily shifted, too. Obviously, I cannot have cusps or corners, or the structure will break.

What I Currently Need:
Advice as to additional functions to graph.
How to use integrals and the disk method for this assignment.

gdbb

You should try to use a different function to define the inner "lining". Something like this:

[tex]a\sqrt{x+b}[/tex]

Also, redefine your sine function to look like this:

[tex]c\sin{(x+d)} + e[/tex]

I messed around with the numbers on WolframAlpha and found that this combination looks decent:

[tex] \frac{1}{4} \sin{x} + 0.3[/tex]

[tex] \frac{2}{5} \sqrt{x} [/tex]

However, I didn't check the integration. I guess you could integrate with the arbitrary constants that I suggested in the functions to try to find a best fit. Might take a lot of work, and might not even work in the end. Best of luck!