Math 3200 (Briggs)
Spring 1998 - Projects

This is a collection of assorted projects that are supported by the material that we will study in Math 3200 this semester. You must complete three (3) projects during the semester (by April 23), and they will each determine 13% of your grade. You may collaborate on the projects, but the final write-up that you submit must be entirely your own work. Your write-ups should be presented neatly with supporting figures, graphs or tables; complete solutions; and discussion of results. You are welcomed to ask me for help or clarification at any time.

Do not base your choice of projects on their apparent length! The projects are designed to require roughly the same amount of time and effort if you are prepared. Some are quite applied and others are more theoretical in nature. The projects are listed more or less in the order in which we will study the relevant material in class. You should be able to get started almost immediately. Do not postpone the projects until the end of the semester. Try to complete approximately one project every 2-3 weeks. Most of all, have fun with them!

Math 3200 - Spring 1998 - Project #1
Draining Reservoirs

Imagine a large water reservoir which loses water due to evaporation. In all that follows, we will let h(t), S(t) and V(t) denote the depth, the surface area and the volume of the water in the reservoir, respectively, at time tex2html_wrap_inline159 . We will always assume that the rate of change of the water volume is proportional to the area of the exposed water in the reservoir; that is, tex2html_wrap_inline161 , where tex2html_wrap_inline163 is a constant (notice the minus sign). In all that follows, we will assume that tex2html_wrap_inline165 meters/day.

  1. First consider a reservoir that has the shape of rectangular prism (parallelepiped) with a constant horizontal cross-sectional area of 200 square meters and a depth of 10 meters (see left figure below).
    1. Verify that the units of tex2html_wrap_inline167 are consistent.
    2. Assuming that the reservoir is filled at t=0, what is the initial volume of the water?
    3. Noting that in this case the surface area of the reservoir is constant, solve the ODE that governs the change in water volume. Use the initial condition to express the volume as a function of time.
    4. At what time (after how many days) will the reservoir be empty, assuming the evaporation rate tex2html_wrap_inline167 remains constant?
  2. Now assume that the reservoir is shaped like an inverted frustrum of a pyramid: an upside-down square-based pyramid whose top has been sliced off (see center figure below). The horizontal cross-sections of the reservoir are always squares which decrease in area from 225 square meters at the top of the reservoir to 100 square meters at the bottom. The depth of the reservoir is 10 meters.

    1. The volume of a pyramid is given by V=1/3 Ah where A is the area of the (square) base and h is the height (see right figure below). Verify that the volume of water in this reservoir when it is full is tex2html_wrap_inline179 cubic meters. (Consider the entire pyramid.)
    2. Again letting h denote the water depth in the reservoir, verify that the surface of the water in the reservoir is a square whose sides have length given by tex2html_wrap_inline183 . (Check that tex2html_wrap_inline185 and tex2html_wrap_inline187 .)
    3. Verify that the surface area of the water is tex2html_wrap_inline189 . (Check that S(0)=100 and S(10)=225.)
    4. Verify that the volume of water in the reservoir is given by tex2html_wrap_inline195 , where tex2html_wrap_inline197 cubic meters. (Check that V(0)=0 and tex2html_wrap_inline201 .) A derivation is required.
    5. Important step: In order to use the ODE tex2html_wrap_inline161 , we must relate the surface area directly to the volume. Show that tex2html_wrap_inline205 .
    6. It now follows that the governing ODE for the volume is tex2html_wrap_inline207 where tex2html_wrap_inline209 .
    7. Solve this ODE (using the initial condition tex2html_wrap_inline211 ) and graph the solution.
    8. At what time (after how many days) will the reservoir be empty, assuming the evaporation rate tex2html_wrap_inline167 remains constant?

INSERT FIGURE

Math 3200 - Spring 1998 - Project #2
Free Fall and Terminal Velocity

We learned in class (see also section 3.4 of the text) that an object in free fall in a gravitational field is governed by the ODE

displaymath131

where m is the mass of the object, g=9.8 meters/sec tex2html_wrap_inline219 is the acceleration of gravity, v(t) is the velocity of the object t seconds after it is released, and tex2html_wrap_inline225 denotes external forces acting on the object. In all that follows, assume that v(0)=0. In this problem, since we will investigate free fall and terminal velocity, let's choose the positive direction for velocity and position as downward, in the same direction as g; therefore, the coefficient of mg in the ODE is +1, not -1.

  1. If there are no external forces acting on the object, then its velocity increases without bound (until the object collides with something). This is unrealistic for motion in the earth's atmosphere, since air resistance is a significant effect. Therefore, assume that air resistance is present and is described by tex2html_wrap_inline237 (k is a constant and the minus sign indicates that the air resistance opposes the motion). Show (include all steps) that the solution of this ODE is

    displaymath132

  2. What is the terminal velocity tex2html_wrap_inline241 of a 100-kilogram object (a small linebacker or a large flower pot) subject to air resistance described by k=5 kg/sec?
  3. Find the function that describes the position x(t) of the object for all tex2html_wrap_inline159 assuming that x=0 corresponds to the position at which the object is dropped.
  4. Make rough sketches of v(t) and x(t).
  5. Now assume that tex2html_wrap_inline255 (this is generally a more accurate way to model air resistance). Solve the resulting ODE for the velocity of the object.
  6. With values of m=100 kg and k=.1 kg/meter, what is the terminal velocity of the object? (Notice that the k's that appear in the two models are different.)
  7. Find the function that describes the position x(t) of the object for all tex2html_wrap_inline159 assuming that x=0 corresponds to the position at which the object is dropped.
  8. Make rough sketches of v(t) and x(t).
  9. In at least two paragraphs, compare and contrast the two models for air resistance.

Math 3200 - Spring 1998 - Project #3
Periodic Drug Doses

Most drugs are eliminated from the body according to a strict exponential decay law. Here are two problems that illustrate the process.

  1. The drug valium has a half-life in the blood (a population average) of 36 hours. Assume that a 50-milligram dose of valium is taken at time t=0. Let m(t) be the amount of drug in the blood in milligrams t hours after the dose. Plot the function m(t) as it varies with time. After how many hours, will the amount of drug reach 10% of its initial value? After how many hours, will the amount of drug reach 1% of its initial value?
  2. Now imagine that a drug (such as aspirin or an antibiotic) with a half-life of 12 hours is taken regularly every eight hours. Assume that the first dose is taken at time t=0. A rough sketch of the amount of drug in the blood is shown in the figure.
    1. What is the amount of drug in the blood at t=8 hours just prior to the second dose?
    2. What is the amount of drug in the blood at t=8 hours just after to the second dose?
    3. What is the amount of drug in the blood at t=16 hours just prior to the third dose?
    4. What is the amount of drug in the blood at t=16 hours just after to the third dose?
    5. Now generalize: what is the amount of drug in the blood at t=8(n-1) hours just prior to the nth dose, where tex2html_wrap_inline295 ?
    6. What is the amount of drug in the blood at t=8(n-1) hours just after to the nth dose?
    7. What can you say about the long-term amount of the drug in the blood? Does it continue to increase without bound or does it approach a steady-state level? If you argue for the latter choice, find the steady-state value of the amount of drug. Justify your conclusion carefully.
    8. Quickly apply the periodic doses problem to the following problem: A fish hatchery harvests 1/3 of its current fish population at the end of each year, and then immediately replenishes the population with 500 new fish. Assuming no deaths and an initial fish population of 1000 fish, what is the steady state population in the hatchery?

INSERT FIGURE

Math 3200 - Spring 1998 - Project #4
Cannon Design

The basic design of a cannon with a stationary carriage and a sliding gun tube-breech block assembly (hereafter called gun assembly) is shown in the figure below. A damping piston between the carriage and the gun assembly absorbs the recoil of the firing. A recoil spring, in a similar position, is designed to push the gun assembly back into the firing position. Assume that the mass of the gun assembly is m kilograms, the recoil spring has a spring constant of k newtons/meter, and the force exerted by the damping mechanism (in newtons) is B times the velocity of the gun assembly. Assume also that when the cannon is fired, an instantaneous velocity of tex2html_wrap_inline307 is imparted to the gun tube.

INSERT FIGURE

  1. Model the horizontal displacement of the gun assembly with a second-order ODE with initial conditions.
  2. Find the solution of the initial value problem with the parameter values m=1500 kg, k=19,500 newton/meter, B=9000 newton-sec/meter, and tex2html_wrap_inline315 meter/sec.
  3. Graph the motion of the gun assembly for the first three seconds after firing.
  4. At what time does the gun assembly first return to the firing position? What is the velocity of the assembly at that time?
  5. At what time does the gun assembly reach it maximum displacement? What is that displacement?

Math 3200 - Spring 1998 - Project #5
Mixing Tank Reactions

  1. A large tank is filled with 500 liters of pure water. At time t=0, an inflow valve is opened and a brine solution with a concentration of 500 grams of salt per liter flows into the tank at 5 liters/minute. At the same time (t=0), an outflow valve is opened and the thoroughly mixed solution in the tank flows out of the tank at 5 liters/minute.
    1. Derive the ODE that describes either the concentration of the solution in the tank or the mass of salt in the tank for all times tex2html_wrap_inline159 .
    2. Solve this ODE together with the appropriate initial condition.
    3. Graph and interpret the solution. In particular, what can you say about the concentration in the tank as tex2html_wrap_inline323 ?
  2. Now imagine that the experiment of part (1) is repeated, except that at t=0, the mixing tank springs a leak and the thoroughly mixed solution also flows out through the leak at a rate of .5 liter/minute.
    1. Derive the ODE that describes either the mass of salt in the tank for all times tex2html_wrap_inline159 .
    2. Solve this ODE together with the appropriate initial condition.
    3. Find the function that gives the concentration of salt in the tank for all times tex2html_wrap_inline159 .
    4. Graph and interpret both the mass and concentration functions.
    5. At what time does the mass function have a maximum?
    6. When does the tank become empty and what is the concentration of the solution in the tank just as the tank becomes empty?

Math 3200 - Spring 1998 - Project #6
Period of the Pendulum

The full equation of motion for the undamped pendulum is

displaymath133

where tex2html_wrap_inline329 , g is the acceleration of gravity, tex2html_wrap_inline333 is the length of the pendulum, and tex2html_wrap_inline335 is measured in radians. (The more familiar linear equation results by assuming that tex2html_wrap_inline337 (small amplitudes) and that tex2html_wrap_inline339 .) An explicit solution of this nonlinear ODE cannot be found in terms of familiar functions. However, it is possible to determine the period of the nonlinear pendulum. Assume the initial conditions tex2html_wrap_inline341 and tex2html_wrap_inline343 . We will consider the quarter period when tex2html_wrap_inline335 decrease from tex2html_wrap_inline347 to tex2html_wrap_inline349 .

  1. What is the period of the linear pendulum (governed by tex2html_wrap_inline351 )?
  2. Now let's work on the full nonlinear ODE. Multiply both sides of the ODE by tex2html_wrap_inline353 , use the chain rule carefully, and apply the initial conditions to show that

    displaymath134

    (Note that tex2html_wrap_inline355 .)

  3. Do you choose the plus or minus branch of the square root? During the quarter period we are considering, is tex2html_wrap_inline357 or is tex2html_wrap_inline359 ?
  4. Now separate variables and write

    displaymath135

  5. The identity tex2html_wrap_inline361 should lead you to

    displaymath136

  6. The function on the right is still difficult to integrate, so we define a new variable tex2html_wrap_inline363 by tex2html_wrap_inline365 . After changing variables you should have

    displaymath137

    where tex2html_wrap_inline367 .

  7. Now we can integrate. Notice that when the original variable tex2html_wrap_inline335 varies from tex2html_wrap_inline347 to tex2html_wrap_inline349 , the new variable tex2html_wrap_inline363 varies from tex2html_wrap_inline377 to tex2html_wrap_inline379 . Letting T be the full period and integrating over a quarter period, show that

    displaymath138

  8. This last integral is called an elliptic integral of the first kind and is denoted as tex2html_wrap_inline383 . Therefore, we have found that the period of the pendulum is tex2html_wrap_inline385 .
  9. What is the period in the limiting case k=0 which corresponds to tex2html_wrap_inline389 .
  10. Use a table of elliptic integrals to find the period of a pendulum with tex2html_wrap_inline391 meters with initial displacements of tex2html_wrap_inline393 .

Math 3200 - Spring 1998 - Project #7
Population Genetics

You may recall from a biology course that the simplest genetic traits are transmitted by a single gene that has one of two forms or alleles. For example, a gene for eye color may be A for the dominant allele (brown eyes) or a for the recessive allele (blue eyes). With one gene coming from each parent, an offspring may have one of three genotypes: AA (brown eyes), Aa (brown eyes), or aa (blue eyes). Suppose that mating takes place randomly (the genes are thoroughly mixed) and that the three genotypes have different fitnesses (or survival probabilities), b, c, and d, respectively. We will let p(t) represent the fraction of A alleles in the gene pool at time t. This means that tex2html_wrap_inline405 and the fraction of a alleles is 1-p. The differential equation that governs how p varies in time is

displaymath139

  1. What are the equilibrium points for this ODE?
  2. Draw the direction field for the case that b = 0.2, c = 0.4 and d = 0.6 and determine the stability of the equilibrium points. Determine tex2html_wrap_inline415 .
  3. Draw the direction field for the case that d = 0.2, c = 0.4 and b = 0.6 and determine the stability of the equilibrium points. Determine tex2html_wrap_inline415 .
  4. Draw the direction field for the case that b = 0.2, d = 0.4 and c = 0.6 and determine the stability of the equilibrium points. Determine tex2html_wrap_inline415 .
  5. Draw the direction field for the case that c = 0.2, b = 0.4 and d = 0.6 and determine the stability of the equilibrium points. Determine tex2html_wrap_inline415 .
  6. Find the general conditions on b, c, and d for polymorphism, the coexistence of both genes in the steady state.

Math 3200 - Spring 1998 - Project #8
A Pursuit Problem

A dog walks north from a crossroads at 1 mile per hour. The dog's master begins one mile east of the crossroads and walks at all times directly at the dog with a speed of s>1 miles per hour.

  1. Find the equation (in the form y=f(x)) that describes the path of the master.
  2. When and where does the master overtake the dog if s = 1.5?
  3. Find the function that gives the meeting time in terms of s.

Math 3200 - Spring 1998 - Project #9
Coding and Information Content

Imagine a coding system in which messages can be formed from a short signal (S) with duration of one time unit and a long signal (L) with a duration of two time units. For example, a message of the form SSLSS, would have a duration of six time units. Let tex2html_wrap_inline443 be the number of different ways of forming a message of length n. For example, tex2html_wrap_inline447 since the messages SSL, LL, SSSS, SLS, and LSS all have a duration of four time units.

  1. What are tex2html_wrap_inline449 and tex2html_wrap_inline451 ?
  2. Show carefully and convincingly that tex2html_wrap_inline453 .
  3. Assume a trial solution of the form tex2html_wrap_inline455 , where tex2html_wrap_inline457 is constant to be determined. Proceed just as you would with a second order constant coefficient ODE and solve this difference equation for tex2html_wrap_inline443 , where tex2html_wrap_inline461
  4. Without computing 100 terms of the sequence tex2html_wrap_inline443 , what is the value of tex2html_wrap_inline465 ?
  5. The capacity of a channel is defined (by Claude Shannon, the innovator of information theory) to be

    displaymath140

    Find the capacity of the channel carrying messages of the form described above.

Math 3200 - Spring 1998 - Project #10
A Predator-Prey Model

The Lotka-Volterra model for describing the interaction between a predator and a prey was formulated in the early part of this century. It has been shown to be fairly accurate when applied to many natural systems (lynx-rabbits, sharks-fish). Let F(t) and R(t) denote the population of a predator and a prey species (think Fox and Rabbit) at time tex2html_wrap_inline159 , measured in hundred of individuals. Consider the predator-prey equations

eqnarray119

  1. First give a brief interpretation of the equations. What is the effect of an increase in the rabbit population on the existing rabbit and fox populations? What is the effect of an increase in the fox population on the existing rabbit and fox populations?
  2. For what fox and rabbit populations is the system at equilibrium?
  3. Sketch the direction field for these equations in the phase plane. (You need to consider only R>0 and F>0. Why?) Choose a few different initial conditions and sketch the resulting trajectories in the phase plane.
  4. Divide the predator ODE by the prey ODE (or vice-versa) to obtain a single ODE in R and F (with t no longer present). Solve this separable ODE to find an implicit representation for the trajectories.
  5. Give the most convincing argument possible that this system has periodic solutions; that is, there is a time interval of length T such that R(t+T)=R(t) and F(t+T)=F(t).
  6. Show that that the average populations are given by R = 1T _0^T R(t) \; dt = 3 and

Bill Briggs
Sun May 17 13:35:03 MDT 1998