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2.3 – Introduction to Mathematical Models 3 phases of mathematical modeling: 1. Formulation – Identify independent and dependent variables, a differential equation describing how these variables interact, and appropriate initial conditions. 2. Solution – Recognizing appropriate analytical (integration) and/or numerical techniques (Euler’s method or direction fields) to use in solving the differential equation. 3. Validation and interpretation – Does the solution make sense? What does it say about the modeled physical phenomenon? Mixture Problems §฀ Problem A tank contains a volume of fluid (such as water in gallons) within which is dissolved a certain amount of solute (such as salt in pounds). Time is measured in minutes. A salt solution enters the tank at a certain inflow rate and the well-stirred (concentration of salt is uniform) solution leaves the tank at a certain outflow rate. How much salt is in the tank at a given time t? The solution is s(t) is the amount of salt in the tank as a function of time. §฀

Model Differential equation: Rate of change of salt in tank = Rate at which salt enters tank – Rate at which salt leaves tank Notation:

V (t ) = c i (t ) = c 0 (t ) = ri (t ) = r0 (t ) =

Rate at which salt enters tank =

Rate at which salt leaves tank

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Rate of change of salt in tank = V(t) is related to flow rates of fluid:

Concentration depends only on time and not on spatial location in the tank. Example1: A) A tank initially contains 1000 gallons of water in which is dissolved 20 pounds of salt. A valve is opened and water containing 0.2 pounds of salt per gallon flows into the tank at a rate of 5 gallons per minute. The mixture in the tank is well stirred and drains from the tank at a rate of 5 gallons per minute. 1. Find the amount of salt in the tank after t minutes. 2. Find the limiting value. Why should such a limit exist? 3. Let the limit in part 2. Be designated as Q L . How long will it be until the amount of salt in the tank is within one percent of Q L ?

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B) A tank initially contains 500 liters of water, in which is dissolved 50 kilograms of glucose. A solution containing 0.4 kilograms per liter of glucose enters the tank at the rate of 2 liters per minute, and the well-stirred mixture leaves the tank at the same rate. 1. Find an expression of the amount of glucose in the tank after t minutes. 2. Is there an equilibrium solution? If so, is it stable? 3. At what time t will there be 100 kilograms of glucose in the container? 4. If we wanted to cut that time in half, what flow rate would be necessary? 5. How would we set up the problem differently if the flow rate was a pulsating 1 + sin(t ) liters per minute? 6. How would we set up the problem differently if the inflow rate was 2 liters per minute but the outflow rate was 3 liters per minute?

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Cooling Problems §฀ Newton’s Law of Cooling The rate of change in the temperature of an object is directly proportional to the difference between its temperature and the temperature of its surrounding medium. §฀

Model Differential equation:

= S (t ) = Assumption: The temperature of the surrounding medium is known for all time of interest and is

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B) A cup of coffee is heated to 140 degrees Celsius and placed to cool in a room where the ambient temperature is 20 degrees Celsius. After 10 minutes, the coffee has cooled down to 90 degrees Celsius, still too hot to drink. How long will it take for the coffee to cool down to a comfortable drinking temperature of 60 degrees Celsius?

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2.4 – Population Dynamics and Radioactive Decay Population Models §฀ Although populations are discrete random variables, differential equation models provide useful approximations when the populations are large and births/deaths occur frequently over the time interval of interest. §฀

Model Differential equation: Rate of change of population = Rate of population increase – Rate of population decrease Two types of rates of change: 1. Percentage (or similar) rates “The population grows at a rate of 6% per year” “Infant mortality is 200 per 100,000 live births” 2. Absolute numbers “Every year the organization admits 10 new members” “Poachers destroy an average of 3000 black bears every month” Notation: P (t )

rb = rd = rb P ( rd P ( M (t

Assumption: The population lives in a well-defined environment called a colony. There are no age, gender, health, or colony location distinctions. The population is sufficiently large.

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Example 1: A) An aquaculture firm raises catfish in ponds. At the beginning of the year, the ponds contain approximately 500,000 catfish. The net growth rate coefficient is estimated to be about 6.1 per 1000 per week. The firm wants to harvest at a constant rate of R fish per week, but it also wants to increase the population to about 600,000 fish by the end of the year. Find the appropriate harvest rate R.

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B) A family-owned farm has a trout pond which initially contains 10,000 trout. The trout population grows at a steady rate of 3% per year, but the family harvests 40 fish per month for their own use. Assuming that this population model does not change, how long will the supply of fish last the family?

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Radioactive De §฀ Differen

Q( t ) = Assumption: No material is added or taken away §฀

Half-life – The length of time it takes for a given amount of a substance to be reduced to one half of its original amount. 1 is derived from the following: Q (t +τ ) = Q (t ),Q (t ) = Ce− kt 2

Example 2: A) Initially, 50 milligrams of a radioactive substance is present. Five days later, the quantity has decreased to 43 milligrams. How much will remain after 30 days?

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B) A radioactive material is known to have a half-life of two weeks. After five weeks, 20 grams of the material is seen to remain. How much material was originally present?

C) Carbon-14, with a half-life of 5730 years, is absorbed by living creatures through respiration and eating, so that the proportion of carbon-14 remains nearly constant throughout the life of an organism. Skeletal remains are discovered by an archaeological team, and carbon-14 testing reveals that only about 30% of the original carbon-14 is still present in the remains. What is the approximate age of the remains?

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