Math 2250 Lab 3 Due Date : 2/2/2017

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1 Math 2250 Lab Due Date : 2/2/2017 Name: UID: Unless stated otherwise, show all your work and explain your reasoning. You are allowed to use any results from lecture or the text as long as they are referenced properly (e.g....by Theorem 1 on page ). You are encouraged to work in groups, but your final written solutions must be in your own words. Make note of your collaborators. Your work must be logically organized and neatly written. (You might want to do a first draft on separate paper.) 1. Brine Tank Modeling The diagrams on the next page represent systems of brine tanks. Each numbered circle corresponds to a brine tank and the value next to each arrow represents a flow rate from one tank to another. For each diagram, derive a system of first order differential equations for the salt amounts x 1 (t), x 2 (t), x (t) and x 4 (t) in the corresponding tanks, measured in kilograms. Assume each tank initially contains the same volume V of liquid, measured in liters L. Flow rates have units of L. min Recall that in a single tank problem where the tank has one inflow and one outflow, then the way to model it is: dx = mass rate in - mass rate out dt dx dt = r in c in r out c out where r stands for the volume flow rate ( L kg ) and c stands for concentration ( ). The min L concentration of salt in the tank changes continuously, but it can be computed by simply dividing the amount of salt in the tank, by the volume of the tank, that is, c out = x(t) kg. V L When you have a system of tanks, each tank may have multiple inflows and or outflows, as in the problems on the next page. Note that both problems are designed so that the total inflow and total outflow rates of each tank are equal and thus the volume of every tank remains constant at V. Hints: Each tank gets its own DE. (So for tank 1 write down a DE for the solute amount x 1 (t), for tank 2 write down a DE for solute amount x 2 (t) etc.) Each arrow going into a tank should correspond with a positive r in c in term in the DE, and each arrow exiting a tank will correspond with a negative r out c out term in the DE. You have not yet learned how to solve a system of differential equations, so there is no need to solve them. Just write each of the differential equations as above. 1

2 (a) (b)

3 2. Consider a bioreactor used by a yogurt factory to grow the bacteria needed to make yogurt. The growth of the bacteria is governed by the logistic equation dp dt = k P (M P ) where P is the population in millions and t is the time in days. Recall that M is the carrying capacity of the reactor and k is a constant that depends on the growth rate. (a) Through obeservation it is found that after a long time the population is the reactor stabilizes at 0 million bacteria, and that when the population of the reactor is 10 million the population increases at a rate of 2 million bacteria per day. From this, find k and M in the governing equation. (b) If the colony starts with a population of 500,000 bacteria, how long will it take for the population to reach 80% of the carrying capacity.

4 (c) Suppose the factory harvests bacteria from the reactor once a week. The harvesting process takes a day, during which the reactor is not operational, leaving 6 days per week for the bacteria to grow in the reactor. The factory wants to maximize the amount of bacteria grown during these 6 days. To achieve this, dp should be at its dt maximum days after harvesting. What initial population (after harvesting) gives the most growth over the 6-day period? What is the population change during that time? (d) Suppose the reactor is modified to allow for continual harvesting without shutting don the reactor. Let h be the rate at which the bacteria are harvested in millions per day. Write down a new differential equation. What is the maximum rate of harvesting h that will not cause the population to go extinct. 4

5 . Phase Diagram Consider the differential equation dx dt = x x 2 x + 1. (a) Find the equilbrium solutions and draw the phase diagram. Classify each equilibrium point as either stable S, unstable U, or semi-stable SS. (b) Use software (e.g. dfield ) to plot the slope field and 5 representative solution graphs, including all equilibrium solution graphs. Set the t axis range to go from -5 to 5 and the x axis range to go from -5 to 5. The graphs should be consistent with your work in part (a). Include your printed plot with this lab. 5

6 4. Terminal Velocity Consider the following linear drag, initial value problem which could arise in studying the velocity of a certain object falling through air, dropped from a high altitude at time t = 0. In this case we have chosen down to be the positive direction, so the acceleration due to gravity is approximately (positive) 9.8 m s 2 : v (t) = v v(0) = 0. (a) Draw the phase diagram for velocity. Find and identify the terminal velocity. (b) Solve the initial value problem. Why is your solution consistent with the phase diagram as t. (c) What percentage of the terminal velocity is v(5)? 6

7 5. (5 points) Parachutist A 100 kg skydiver drops from a hot air balloon. After 5 seconds of free fall, a parachute is opened. The parachute immediately introduces a drag force proportional to the velocity. After an additional 6 seconds, the parachutist reaches the ground. Assume that air resistance is negligible during free fall and that the parachute is designed so that a 100 kg person will reach a terminal velocity of 8 m/s. (g = 9.8 m/s 2 ) (a) What is the speed of the skydiver immediately before the parachute is opened? (b) What is the parachutist impact velocity? Give your result to 2 decimal places. Do you think a different parachute should be designed? (c) At what altitude was the parachute opened? Give your result to 2 decimal places. (d) What is the balloon altitude? Give your result to 2 decimal places. 7

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