Department of Mathematics

Math 300: Mathematical Computing

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Assignment 9

Your assignment is to find the volume of a wine fermentation tank made by Sonoma Stone. When converted to the metric system (because wine bottles have a volume of 750 milliliters), the boundary of the central cross section of the interior of the tank is very nearly $$0.0002019698840x^2+0.0001341299036\frac{y^2}{0.9701781639+\alpha y}=1.$$ All distances are measured in centimeters. In other words, the interior of the tank is interior of the surface described by rotating that curve about the $y$ axis. The volume of that region can be found as an integral. I.e. we are evaluating an integral to find the volume of a surface of rotation. For the real tank, $\alpha \approx 0.004$.

1. Create a Maple document (or a Maple command file, if you prefer) to evaluate the volume for any value of $\alpha$ between 0 and 0.02. Comment the document liberally.
2. Use the implicitplot and implicitplot3d commands to plot both the cross section of the tank, and a 3-d view of the interior of the tank, when $\alpha =0.004$.
3. Plot the value of the volume of the tank for values of $\alpha$ between 0 and 0.02.
4. Write a LaTeX document describing how to find the volume of the tank. Remember that this is a mathematical discussion - you will use mathematical notation to describe the process - not computer or Maple notation. Moreover, the instructor has no interest in a description of the computational steps in finding the volume - he wants to know how you solved mathematically for the various components of the integral. Obviously, your document will include some astute observations about the way that the volume depends on $\alpha$, and so on.
5. Make sure to tell how many bottles of wine one expects to get out of the tank when $\alpha =0.004$. Do not try to include the solution for all values of $\alpha$, unless it were in an appendix.
6. Make sure to include the pictures of the cross-section and 3-d view of the tank that you created, as well as the one for variation of volume with $\alpha$.

This assignment is worth 60 points and is due at 3:00 on Friday, 7 December (a day that will live in infamy). It is turned in when a message containing three attachments appears in the instructor's email inbox. The attachments are a) the Maple .mw file (or if you did a .mc file, send that); b) the .tex file; and c) the .pdf file you created from the LaTeX .

Welcome to Math 300 for summer, 2013.