Maths Expert Needed. 20 Hours. Original Work Required

The majority of the credit you receive will be based on the clarity of your ability to express your written answers and your work shown.

Project Description:

Mathematical modeling can often include the process of writing a differential equation to describe a physical situation. Almost all of the differential equations that you will use in your future job are there because at some time someone modeled a situation to come up with the differential equation that you are using. So far, you have studied the graphs for differential equations graphically and learned some analytically techniques for solving certain types of differential equations: separable and linear first order. This project is designed for you to utilize these skills by focusing on the questions below. You will turn in, on BlackBoard, solutions to these problems that include explanations where applicable.

Many important problems in biology and engineering can be put into the following framework: A solution containing a fixed concentration of substance x flows into a tank, or compartment, containing the substance x and perhaps some other substances, at a specific rate. The mixture is stirred together very rapidly, and then leaves the tank, again at a specific rate. Find the concentration of substance x in the tank at any time t. Problems of this type fall under the general heading of “mixing problems” or compartment analysis. This project explores this application.

Independent Tanks

1. Create models for the following scenarios:

(a) A very large tank initially contains 15 gallons of saltwater containing 6 pounds of salt. Saltwater containing 1 pound of salt per gallon is pumped into the top of the tank at a rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute. Call this tank “Tank A”.

(b) Tank B is the same basic scenario as Tank A, but pure water is being pumped into Tank B instead of saltwater.

(c) Tank C is the same basic scenario as Tank A, but the rates are switched: saltwater enters Tank C at a rate of 1 gallon per minute, and leaves at a rate of 2 gallons per minute.

(d) Tank D is the same basic scenario as Tank A, but Tank D initially contains 6 gallons of pure water.

2. Graph the slope fields for each of the differential equations described in part (1). What do you predict is the long term behavior of the amount of salt in each tank for different initial conditions?

3. Solve initial value problems that correspond to individual Tanks A, B, C, and D. Use a plot to compare four solution curves and discuss how these curves predict/represent outcomes you predicted from the description of each scenario and the graphs in part (2).

Dependent Tanks

You have been given a pair of large tanks with saline solution. Tank E began with 3 pounds of salt in 14 gallons of water. A saline mixture containing 4 pounds of salt per gallon flows into tank E at a rate of 2

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MATH 310

Math 310 Project 1 Overview DUE: February 27, 2018 at 11:59pm

gallons per minute. The tank is kept well mixed, and flows out of tank E into tank F at a rate of 2 gallons per minute. Tank F initially contained 31 gallons of fresh water, and the tank is kept well mixed, and that mixture if pumped out at a rate of 2 gallons per minute.

1. Using the above description, write down a differential equation, with an initial condition, for the amount of salt in tank E. Use sE to represent the salt in tank E.

2. Write down a differential equation, with an initial condition, that gives the amount of salt in tank F. Use sF to represent the salt in tank F .

3. After the tanks have been mixing for one hour, a leak is sprung in tank F, so that an additional 1 gal/minute is flowing out of the tank. Call the time that the leak sprang time t = 0. For the sake of this problem, we will relabel this leaky tank as tank G. Let sG be the salt in tank G. Write a differential equation, that describes the amount of salt in tank G. What is the appropriate initial condition for tank sG? Explain your answer.

Real World Application

Many real world problems can be thought of as a “mixing problem”, i.e. the analysis of a single “compart- ment” into which some substance is flowing at a certain rate and out of which the same substance is flowing at some, probably different, rate. Examples of such situations include: chemical mixtures, pollution, gaseous mixtures and even some organ functions. Research and discuss, in a few paragraphs, a “real life” scenario that can be modeled as a mixing problem.

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