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Math 220 1.3 Differential Equations as Mathematical Models (1st Order) 1. Radioactive Decay: Suppose it is assumed that a radioactive substance decays at a rate that is proportional to the amount of the substance (A) present at time, t. Determine a differential equation for the given scenario. 2. Malthusian Population Model: In the late 18th century, the economist Thomas Malthus constructed one of the earliest models for population growth. The assumption behind the model is that the rate at which the population of a region grows is proportional to the size of the total population at that time. a. Determine a differential equation for the population, P(t). b. Assume that the constant of proportionality is equal to 0.02. Using Maple, sketch the slope field for the Malthusian Population Model. What do we expect will happen to the population as t ? Is this realistic? Explain. 3. Newton’s Law of Cooling/Warming: Experiments have shown that given certain conditions, a good approximation for the temperature of an object can be obtained by using Newton’s law of cooling/warming: The temperature of a body changes at a rate that is proportional to the difference between the temperature of outside medium ( Tm ) and the temperature of the body itself (T). a. Express Newton’s Law of Cooling/Warming as a differential equation. b. Suppose a chicken is taken out of the freezer and placed on the kitchen table. What is a reasonable assumption for the initial temperature of the chicken? The temperature of the surrounding medium? c. Using the information from part b and assuming that the constant of proportionality is equal to -.025, set up a specific differential equation for the temperature of the chicken. Using Maple, produce a sketch of the slope field for the DE. Have Maple sketch the particular solution based on the stated initial condition. Explain what happens to the temperature of the chicken as time passes. 4. Falling Bodies: According to Newton’s 2nd Law of Motion, when a net force acting on a body is not zero, then the net force is proportional to the acceleration; specifically F ma . If we assume that the only force acting on the falling body is gravity (i.e. we neglect air resistance), what is the differential equation for the velocity of the body at time, t? What is the differential equation related to the position of the body at time, t? (Let a positive sign represent upward direction and negative sign represent downward direction). 5. Falling Bodies with Air Resistance: Using the model from #4, we will now take the force of air resistance into account. Assuming that the air resistance force for a falling body with mass, m, is proportional to the body’s instantaneous velocity and acts in direct opposition to the force of gravity, set up a differential equation for the object’s velocity. Also, set up a DE for the position of the body at time, t. (For this example, let a positive sign represent downward direction and a negative sign represent upward direction). 6. Series Circuits: According to Kirchhoff’s Second Law (for voltage) the impressed voltage E(t) on a closed loop circuit is equal to the sum of the voltage drops across the circuit. The voltage drops are listed below for the components contained in a(n) RLC circuit: Component Voltage Drop Inductor (L) di L dt Resistor (R) iR Capacitor 1 q (C) C With i representing current and q representing charge, the relationship between them is dq described by i . dt a. Determine the differential equation for an RL circuit in terms of the current, i(t). b. Determine the differential equation for an RLC circuit in terms of the charge, q(t). 7. Mixtures Suppose a salt solution (mixture of water and salt, frequently referred to as “brine”) is pumped into a tank already holding fluid, is well stirred and then pumped out of the tank. If we let A = the amount of salt in the tank at time, t, then the rate of change of salt in the tank at time, t, can be expressed as: dA input rate output rate Rin Rout dt of salt of salt We usually know the concentration of the salt and the flow rate of the fluid. In general, how would we determine the input and output rates of salt if given the concentration of salt and the flow rate of the solution? Consider the following scenarios: a) A mixing tank initially holds 300 gallons of brine (salt solution). A salt solution is pumped into the tank at a rate of 3 gal/min. The concentration of the salt in the mixture coming into the tank is 2 lbs/gal. The solution is stirred and then pumped out at the same rate as the solution flowing in. Determine a DE to model the given scenario. b) 50 gallons of brine initially containing 20 lbs of salt are in a tank into which 2 gallons of pure water run each minute with the same amount of mixture flowing out each minute. Determine a DE for modeling the rate of change of the salt.