Practice Problems For Test 1

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1 Practice Problems For Test 1 Population Models Exponential or Natural Growth Equation 1. According to data listed at the world s total population reached 6 billion persons in mid-1999, and then increasing at the rate of about 212 thousand persons each da. Assuming that natural population growth at this rate continues, we want to answer these questions: (a) What is the annual growth rate k? (b) What will be the world population at the middle of the 21st centur? (c) How long will it take the world population to increase tenfold - thereb reaching the 60 billion that some demographers believe to be the maximum for which the planet can provide adequate food supplies? 2. A bacteria culture grows at a rate proportional to the amount of bacteria present. If a patient is initiall infected with 400 bacteria and the bacteria population reaches 1400 in 2 hours, how long will it take for the bacteria population to reach a critical level of 2000 bacteria? 3. In a certain culture of bacteria, the number of bacteria increased sixfold in 10 hours. How long did it take for the population to double? 4. A certain cit had a population of 25,000 in 1960 and a population of 30,000 in Assume that its population will continue to grow exponentiall at a constant rate. What population can its cit planners expect in the ear 2000? General Population Equation 1. Suppose that an alligator population numbers 100 initiall, and that its death rate is δ = 0 (so none of the alligators is ing). Assume the birth rate is β = (0.0005)P and thus increases as the population does. Set up the general population equation and solve the initial value problem. What happens to the population after twent ears? Extinction-Explosion Equation 1. Consider an animal population P(t) that is modeled b the equation dp = P(P 150) Solve the equation and find particular solutions for the following initial values: (a) P(0) = 200 (b) P(0) = Consider an animal population P(t) with constant death rate δ = 0.01 (deaths per animal per month) and with birth rate β proportional to P. Suppose that P(0) = 200 and P (0) = 2. (a) When is P = 1000? (b) When does doomsda occur? Logistic Equation (Bounded Populations) 1. A certain population can be divided into two parts, those who have are infected b a disease and those who are not infected but who are susceptible. If represents the proportion of people in the population who have the disease, and (1 ) the proportional of people who are susceptible, then the initial value problem that models the proportion of infected people is = α(1 ), (0) = 0 (a) Solve the initial value problem. (b) If 0 < 0 < 1, determine the value of lim t (t). 2. Suppose that a communit contains 15,000 people who are susceptible to Michaud s sndrome, a contagious disease. At time t = 0 the number N(t) of people who have developed Michaud s sndrome is 5,000 and is increasing at the rate of 500 per da. Assume that N (t) is proportional to the product of the numbers of those who have caught the disease and of those who have not. How long will it take for another 5,000 people to develop Michaud s sndrome? 1

2 Differential Equations - Homework For Test The logistic initial value problem: (a) Find the solution dp = kp(m P), P(0) = P 0 (b) Use our solution to find the solution to the word problem: Suppose that at time t = 0, 10 thousand people in a cit with population 100 thousand people have heard a certain rumor. After 1 week the number P(t) of those who have heard it has increased to P(1) = 20 thousand. Assuming that P(t) satisfies a logistic equation, when will 80% of the cit s population have heard the rumor? 4. Find the inflection point for the logistic equation: Equilibrium Solutions and Stabilit dp = kp(m P) 1. For each of the following, determine the equilibrium point(s) and classif as stable, semi-stable, or unstable. Draw several graphs of solutions in the tx plane. You do not need to solve the equations. (a) = x 4 (b) = x2 4 (c) = x2 5x + 4 (d) (e) = (2 x)(6 x) = x(1 x)2 (f) = e x 1 2. A fish population can be modeled b the logistics equation (1 = k ), (0) = 0 > 0 where k and M are positive M constants. (a) Sketch several solution curves for various values of 0. (b) Based on our graph, what is the phsical meaning of M? Harvesting a Logistic Population 1. To model a fish population in which large scale fishing occurs, we can use the equation (1 = k ) H(), (0) = 0 > 0 M where k and M are positive constants and H() represents the fish captured. Note that M has the same meaning as in the previous problem. (a) Suppose H() = 0.1. Describe what this means in terms of the fish population captured. (b) Suppose H() = 0.1M. Describe what this means in terms of the fish population captured. (c) Consider the StreamPlot for the specific example ( = ) h. Enter the following Mathematica code 100 and describe what happens when the number of fish harvested (h) is graduall increased from 0 to 2. Manipulate [ StreamPlot [ { 1, 0. 1 (1 /100) h}, { t, 0, }, {, 10,120}], {h, 0, 5 } ] (d) If ou move the slider to a larger values of h, ou eventuall have onl one stabilit point. Determine the exact value of this value of h for the equation (1 = k ) h. M Bifurcation Diagrams 1. Create a bifurcation diagram for the following differential equations: (a) = x(4 x) h (b) = 1 x(10 x) h 10 (c) = kx x3 (d) = x + kx3

3 Differential Equations - Homework For Test 1 3 Interactions of Logistic Populations For each two-populaiton sstem first describe the tpe of x- and -populations involved (exponential or logistic) and the nature of their interaction - competition, coorperation, or predation. Then find and characterize the sstem s critical points (as to tpe and stabilit). Determine what nonzero x- and -populations can coexist. Finall, construct a phase plane potrait that enables ou to dexribe the long-term behavior of the two populations in terms of their initial populations x(0) and (0) = 2x x = 2x 4x, = 2x 16x, = 3 x = 3x x2 1 2 x, = 3x x x, = 3x x2 1 4 x, = 30x 3x2 + x, = 30x 2x2 x, = 30x 2x2 x, = x 3 = 4 x = 4 2x = 1 5 x = x 2 Newton s Law of Cooling = x = x = x 1. A pitcher of buttermilk initiall at 25 C is to be cooled b setting it on the front porch, where the temperature is 0 C. Suppose that the temperature of the buttermilk has dropped to 15 C after 20 min. When will it be at 5 C? 2. A cup of coffee is moved from the microwave to our desk when the coffee has a temperature of 200 F. The room temperature is 63 F and the temperature of coffee is 70 F after 25 minutes. If ou like to drink coffee at a temperature of 170 F, when should ou drink the coffee? 3. A cake is removed from an oven at 210 F and left to cool at room temperature, which is 70. After 30 min the temperature of the cake is 140 F. When will it be 100 F? 4. Just before midda the bo of an apparent homicide victim is found in a room that is kept at a constant temperature of 70 F. At 12 noon the temperature of the bo is 80 F and at 1 pm it is 75 F. Assume that the temperature of the bo at the time of death was 98.6 F and that it has cooled in accord with Newton s law. What was the time of death? 5. A hit-and-run accident occurs at 10:00 P.M. and a suspect is arrested 2 hours later with a blood alcohol level of 0.07%. After waiting an additional three hours, a second blood alcohol reading is found to be 0.06%. Assuming that the amount of alcohol in the blood decreases exponentiall, what was the blood alcohol level at the time of the accident? 6. Consider a heating and cooling problem given b the equation T = temperature, in F dt = k(t M) + H(t) where M = surrounding temperature k = a constant t = time, in hours H(t) = heating or cooling source A building has a solar hearing sstem that consists of a solar panel and a hot water tank. The tank is well-insulated and has a constant of k = 1/50. Under sunlight energ generated b the solar panel will increase the water temperature in the tank at the rate of 2 F/hour. Suppose at 9:00 A.M. the water temperature is 100 F and the room temperature where the tank is stored is a constant 70 F. Find the temperature of the water in the tank at an time t.

4 Differential Equations - Homework For Test 1 4 Torricelli s Law 1. A spherical tank of radius 4 ft is full of gasoline when a circular bottom hole with radius 1 in. is opened. How long will be required for all the gasoline to drain from the tank? 2. A water tank has the shape obtained b revolving the parabola x 2 = b around the -axis. The water depth is 4 ft at 12 noon, when a circular plug in the bottom of the tank is removed. At 1 pm the depth of the water is 1 ft. (a) Find the depth (t) of water remaining after t hours. (b) When will the tank be empt? (c) If the initial radius of the top surface of the water is 2 ft, what is the radius of the circular hole in the bottom? 3. A hemispherical tank with radius R is initiall full of water. The tank has a circular hole of radius r in the bottom of the tank. (a) Show that the differential equation that describes the volume V of water in the tank is dv = π 0.01πh2 (b) Using the formula for the volume of a cone (V = 1 3 πr2 h) to find an expression of the form t = f (h) that gives the time it takes for the water level to reach a height h. x Natural Deca Models x 2 + ( R) 2 = R 2 (a) Show that the differential equation that describes the height of the water in the tank is = r2 2g x 2 (b) Assuming (0) = R, find an expression for the time it takes to empt the tank. 4. A conical tank 15 feet deep with an open top has a radius of 15 feet. Initiall the tank is empt but water is added at π ft 3 /hr. Water evaporates from the tank at a rate 0.01 times the surface area of the water. x 1. The half-life of radium-226 is 1620 ears. Find the time when the initial amount of material is reduced to one-tenth of its original amount. 2. An accident at a nuclear power plant has left the surrounding area polluted with radioactive material that decas naturall. The initial amount of radioactive material present is 15 su (safe units), and 5 months later it is still 10 su. (a) Write a formula giving the amount A(t) of radioactive material (in su) remaining after t months. (b) What amount of radioactive material will remain after 8 months? (c) How long - total number of months or fraction thereof - will it be until A = 1 su, so it is safe for people to return to the area? 3. A certain moon rock was found to contain equal numbers of potassium and argon atoms. Assume that all the argon is the result of radioactive deca of potassium(its half-life is about ears) and that one of ever nine potassium atom disintegrations ields an argon atom. What is the age of the rock, measured from the time it contained onl potassium?

5 Differential Equations - Homework For Test 1 5 Mixture Problems 1. A tank contains 1000 liters (L) of a solution consisting of 100 kg of salt dissolved in water. Pure water is pumped into the tank at thre rate of 5 L/s, and the mixture - kept uniform b stirring - is pumped out at the same rate. How long will it be until onl 10 kg of salt remains in the tank? 2. A tank initiall contains 60 gal of pure water. Brine containng 1 lb of salt per gallon enters the tank at 2 gal/min, and the (perfectl mixed) solutions leaves the tank at 3 gal/min; thus the tank is empt after exactl 1 hour. (a) Find the amount of salt in the tank after t minutes (b) What is the maximum amount of salt ever in the tank? 3. A 400-gal tank initall conatins 100 gal of brine containing 50 lb of salt. Brine containg 1 lb of salt per gallon enters the tank at the rate of 5 gal/s, and the well-mixed brine in the tank flows out at the rate of 3 gal/s. How much salt will the tank contain when it is full of brine? 4. Consider the cascade of two tanks with V 1 = 100 (gal) and V 2 = 200 (gal) - the volumes of brine in the two tanks. Each tank also initiall contains 50 lb of salt. The three flow rates are each 5 gal/min, with pure water flowing into tank 1. (a) Find the amount x(t) of salt in tank 1 at time t. (b) Find the amount (t) of salt in tank 2 at time t. (c) Find the maximum amount of salt ever in tank A 2 liter beaker initiall contains 100 ml of pure water. A solution containing 0.3 grams/liter of copper sulphate is added to the beaker at a rate 25 ml/min. The well mixed solution is drained off at the same rate. (a) Write the initial value problem describing the amount of copper sulphate in the beaker at an time t. Use the initial value problem to predict the amount of copper sulphate in solution as t. (b) Sketch the StreamPlot of our model (c) Solve the initial value problem and determine the amount of copper sulphate in the beaker after 2 hours. 6. A 2 liter beaker initiall contains 100 ml of pure water. A solution containing 0.3 grams/liter of copper sulphate is added to the beaker at a rate of 25 ml/min. The well mixed solution is drained off at the rate 20 ml/min. (a) How long will it take for the beaker to be completel full? (b) How much copper sulphate is in the beaker when the beaker is full? 7. Into a tank containing 40 gallons of fresh water, Julie was supposed to add 0.5 pounds of acid but but accidentl added 1.5 pounds of acid. To correct her mistake, she started adding fresh water at a rate of 3 gal/min while draining off the mixture at the same rate. How long will it be until the tank has the correct amount of acid? Theor Problems For Test 1 1. Verif that = 2x3 + C 3x 2. For (x) = 1 x is a solution to x + = 2x 2. (a) Solve the above differential equation with the initial value (0) = 1 2. (b) Use Mathematica to draw a StreamPlot of the differential equation and the solution to the initial value problem in the region 2 x 2, 2 2. As an example, the initial value problem = cos x, (0) = 1 has the solution = sin x + 1. The StreamPlot and solution curve in the region 6 x 6, 4 4 can be drawn using the following code. p1=streamplot [ { 1, Cos [ x ] }, { x, 6,6},{, 4, 4 } ] ; p2=plot [ Sin [ x ]+1,{ x, 6,6}, P l o t S t l e >{Thickness [ ], Red } ] ; Show[ p1, p2 ] The input for StreamPlot is the derivative term, written as a vector. So a differential equation x = would be sin x + cos entered as

6 Differential Equations - Homework For Test 1 6 StreamPlot [{ Sin [ x ]+Cos [ ], x },... ] 3. A particle starts from the origin and has velocit v(t) described b the diagram. Sketch the graph of x(t), the position of the particle and an time t. 2 1 v(t) B hand, on a piece of graph paper, sketch the slope field for the differential equation = x at the points (0, 4), (2, 2), (4, 0), (2, 2), (0, 2), ( 2, 2), ( 2, 0), and ( 2, 2). Based on the graph, guess the solution to the initial value problem = x, (0) = Consider the StreamPlot for the differential equation = x on the interval 2 x 2, 2 2. (a) Guess the equation of the solution curve through the point (0, 1). (b) Explain wh ou cannot find the equation of the solution curve through (0, 0). Could this have been predicted from the equation? 6. Consider the differential equation = 9 2. (a) Use Mathematica to draw the StreamPlot on the region 3 x 3, 4 4. (b) Suppose 1 (x) is the solution curve that passes through the point (0, 4) and 2 (x) is the solution that passes through (0, 2). Draw 1 (x) and 2 (x) on the graph. (c) Explain wh lim x 1 (x) = lim x 2 (x). 7. For the initial value problem x 2 = 0, (0) = 0 t (a) Use Mathematica to draw the StreamPlot on the region 1 x 1, 1 1. (b) Show that = x is a solution. 4 cx (c) Show the = 0 is a solution. (d) Does this violate the Existence and Uniqueness Theorem? Explain wh or wh not. 8. For the following differential equation: (a) Find a general solution. = 2 (b) Determine a value of C so that (10) = 10 (c) Is there a value of C such that (0) = 0? (d) Can ou find b inspection a soluiton of = 2 (0) = 0? (e) Create a direction field. Can ou conclude that, given an point (a, b) in the plane, the differential equation has exactl one solution satisfing (a) = b? 9. For the following initial value problem: = 1 x (a) Create a slope field of the differential equation. (b) Solve the initial value problem. (0) = 0 (c) Do our results from part b match our direction field from part a? (d) Can ou find all points (a, b) that lead to no solution to the initial value problem = 1 x (a) = b 10. Find all solutions to the initial value problem: = 2 (0) = For the following initial value problem: x = 2 (a) = b (a) Find all points (a, b) that lead to no solution for the initial value problem. (b) Find all points (a, b) that lead to a unique solution for the initial value problem. (c) Find all points (a, b) that lead to an infinite number of solutions for the initial value problem.

7 Differential Equations - Homework For Test For the following initial value problem: x = 0 (a) = b (a) Solve the differential equation. (b) Find all points (a, b) that lead to no solution for the initial value problem. (c) Find all points (a, b) that lead to a unique solution for the initial value problem. (d) Find all points (a, b) that lead to an infinite number of solutions for the initial value problem. 13. For the following initial value problem: = 3 2/3 (a) = b (a) Solve the differential equation. (b) Find all points (a, b) that lead to no solution for the initial value problem. (c) Find all points (a, b) that lead to a unique solution for the initial value problem. (d) Find all points (a, b) that lead to an infinite number of solutions for the initial value problem. Problems For Test 1: Answers Exponential or Natural Growth Equation 1. (a) k (b) P(t) = 6e.0129t (c) t 178 Year t 2.57 hours 3. t 3.87 hours 4. 51,840 P(51) billion General Population Equation 1. P(t) = t Population Explosion as t 20 Extinction-Explosion Equation 1. (a) P(t) = e.06t (b) P(t) = e.06t 2. P(t) = e.01t (a) t months (b) t months Logistic Equation 1. (a) (t) = (b) lim t (t) = 1 0 e αt + 0 (1 e αt ) 2. N(t) = e.15t t 9.24 das 3. (a) P(t) = (b) P(t) = MP 0 P 0 + (M P 0 )e kmt e.8109t t 4.42 weeks 4. P = k 2 P(M P)(M 2P) P = 0 P = M P = M/2 Equilibrium Solutions and Stabilit 1. (a) x = 4 unstable (b) x = 2 stable x = 2 unstable (c) x = 1 stable x = 4 unstable (d) x = 2 stable x = 6 unstable (e) x = 0 unstable x = 1 semi-stable (f) x = 0 stable 2. M is the carring capacit: maximum population the environment can support

8 Differential Equations - Homework For Test 1 8 Harvesting a Logistic Populations 1. (a) We harvest 10% of the fish population per time interval (b) We harvest 10% of the limiting population (c) streamplot (d) h = km 4 = (0.1)(100) = 2.5 Equilibrium Solution = 50 4 Bifurcation Diagrams 1. (a) h = 4 (b) h = 2.5 (c) k = 0 (d) k = 0 Interactions of Logistic Populations 1. exp / exp / competition 2. exp / exp / cooperation 3. exp / exp / predation (x: predator : pre) 4. log / exp / competition 5. log / exp / cooperation 6. log / exp / predation (x: pre : predator) 7. log / log / cooperation 8. log / log / predation (x: pre : predator) 9. log / log / predation (x: pre Y: predator) Newton s Law of Cooling 1. t 63 minutes 2. t 2 minutes 3. t minutes 4. Approximate time of death is 10:30 am % 6. T(t) = Ce 1 50 t Torricelli s Law 1. t minutes 2. (a) 3/2 = 7t (a) (b) t minutes 21 (c) r = 30 inches (b) 4 3 R3/ /2 = r 2 2g t R5/2 4. (a) (b) t = f (h) = 500 ln 1.1h 1 +.1h Natural Deca Models 1. t ears 2. (a) A(t) = 15e t 5 ln 2 3 (b) A(8) 7.84 su (c) t months 3. 4,250,000,000 ears old Mixture Problems 1. t 7.68 minutes 2. (a) x(t) = 60 t 1 (60 t)3 602 (b) x 23.1 lbs 3. x lbs 4. (a) x(t) = 50e 1 20 t (b) (t) = 150e 1 40 t 100e 1 20 t (c) t minutes 5. (a) x +.25x =.0075 x(0) = 0 (b)

9 Differential Equations - Homework For Test 1 9 (c) x(t) =.03(1 e.25t ) x(120) (a) t 6.3 hours (b).600 grams 7. t 22 seconds