ode_projects

# ode_projects - Math 3200 Mini-Projects(modied from Bill...

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Math 3200 Mini-Projects (modified from Bill Briggs 2004) This collection of assorted mini-projects is supported by the material that we will study this semester. You must complete two (2) mini-projects during the semester. The first mini-project is due no later than March 4 and must be selected from projects 1–4. The second mini-project is due no later than May 7 and must be selected from projects 5–10. Do not procrastinate; there will be no extensions! Each project will determine 7% of your grade. You may collaborate on the projects, but the your final write-up must be entirely your own work . The solutions to the mini-projects must be neat, legible, and written with perfect spelling and grammar. It would be best to use a word processor or L A T E X to write at least the text portions of every assignment. If you do write by hand, it must be effortless to read. A typical solution should consist of a statement or summary of the problem; a description of the overall approach to the problem; a well-justified solution; a presentation of results including relevant tables and graphs; and a discussion that includes an interpretation of the solution, an assessment of accuracy, and/or other interesting observations. Do not base your choice of projects on their apparent length! The projects are designed to require roughly the same amount of time and effort if you are prepared. Some are quite applied and others are more theoretical in nature. The projects are listed more or less in the order in which we will study the relevant material in class. You should be able to get started almost immediately. If you want to work on a project that requires material we have not yet covered in class, I’ll be glad to help you get started. I am always available for questions and consultation on the projects. Most of all, have fun with them! 1. Evaporating Reservoirs . Imagine a large water reservoir that loses water due to evaporation. In all that follows, we will let h ( t ), S ( t ) and V ( t ) denote the depth, the surface area, and the volume of the water in the reservoir, respectively, at time t 0. We will always assume that the rate of change of the water volume is proportional to the surface area of the exposed water in the reservoir; that is, V ( t ) = αS ( t ), where α = 0 . 05 meters/day (notice the minus sign). (a) First (warm-up) consider a reservoir that has the shape of rectangular prism (a box or parallelepiped) with a constant horizontal cross-sectional area of 200 square meters and a depth of 10 meters (see left figure below). i. Verify that the units of α are consistent. ii. Assuming that the reservoir is filled at t = 0, what is the initial volume of the water? iii. Noting that in this case the surface area of the reservoir is constant, solve the ODE that governs the change in water volume. Use the initial condition to express the volume as a function of time.

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