Problem of the Month for March, 2009

Problem: A thin and somewhat porous container in the shape of an inverted cone has a height of 16 cm and a radius of 5 cm at the top. It is partially filled with a liquid that oozes out through the sides at a rate that is proportional to the area of the container that is in contact with the liquid. If we pour the liquid into the container at a rate of 2 cubic cm per minute, then the height of the liquid decreases at a constant rate of 0.3 cm per minute when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour liquid into the container?

Possible Extensions/Generalizations to Consider: You might try considering the general case where the height of the container is H, the radius of the container is R, when we pour liquid into the container at a rate of a cubic cm per minute, then the height of the liquid decreases (or perhaps increases) at a constant rate of b cm per minute when the height is c cm. If our goal is to keep the liquid at a constant height of c cm, at what rate should we pour liquid into the container? Your answer here would, of course, depend on all the variables. You could consider containers shaped in other ways: an upside-down pyramid, a bowl, etc…

Rules

1. You must be a Bethel University student during the given month.
2. Your solution should be written in complete sentences (at least if you want to win) and either typed or written very neatly by hand. Equations and diagrams may be included by hand or by computer as necessary.
3. Your solution must be turned in to P.O. 95 by 4 PM on the last day of classes of the given month.
4. The winner will be the person who does the best job answering the problem as judged by a faculty member of the math and computer science department. If more than one person answers the problem correctly, the person who does the best job in communicating their solution and/or considering generalizations of the given problem will win. If no one answers the problem correctly, the best attempt will win.
5. Do not put your name on your solution paper. Instead, put your Bethel ID number in the top right corner of your solution paper.

Suggestions

1. Be thorough, yet concise. Be sure to answer the question completely and in such a way that clearly communicates your solution, while at the same time being as efficient in your communication as possible.
2. If you think other people will also answer the question correctly and are also good writers, you can increase your chances of winning by considering and writing about possible generalizations of the given problem and the solutions to those generalizations. However, a correct answer to the original problem that does not consider generalizations will beat out an incorrect answer to the original problem that does consider generalizations. In short, make sure your answer to the original problem is correct before considering any generalizations.
3. Neatness counts. Grammar and spelling count. When relevant, pictures are helpful.
4. Explicitly state any assumptions you are making. If you are unsure whether a particular assumption is “allowed”, say so in your write-up but then answer the question by either making the assumption in question or by stating why you think you can’t or shouldn’t make the assumption.

Prizes and Benefits

1. Your picture and a short biographical sketch will be posted, as well as your solution, for all to admire. This will be done temporarily in the math and computer science hallway, and, perhaps, for as long as Bethel exists on the internet. You can inspire and show your accomplishment to your friends, children, grandchildren, your future bosses, and more!
2. You will win a $25 gift certificate from House of Wong restaurant.
3. You will earn some extra credit in your math and computer science courses of the given month (amount to be determined by your professor).
4. You might be able to get an extension on an assignment for your math/cs courses if you are working on the problem of the month (discuss this with your professor).