Chapter 1

Two kilos of a fish poison, rotenone, are mixed into a lake which has a volume of \(100 \times 20 \times 2=4000\) cubic meters. No water flows into or out of the lake. Fifteen percent of the rotenone decomposes each day. a. Write a mathematical model that describes the daily change in the amount of rotenone in the lake. b. Let \(R_{0}, P_{1}, R_{2}, \cdots\) denote the amounts of rotenone in the lake, \(P_{t}\) being the amount of poison in the lake at the beginning of the \(t \underline{h}\) day after the rotenone is administered. Write a dynamic equation representative of the mathematical model. c. What is \(R_{0} ?\) Compute \(R_{1}\) from your dynamic equation. Compute \(R_{2}\) from your dynamic equation. d. Find a solution equation for your dynamic equation.

Write a solution equation for the following initial conditions and difference equations or iteration equations. In each case, compute \(B_{100}\). a. \(\quad B_{0}=100 \quad B_{t+1}-B_{t}=0.2 B_{t}+5\) b. \(\quad B_{0}=138 \quad B_{t+1}-B_{t}=0.05 B_{t}+10\) c. \(\quad B_{0}=138 \quad B_{t+1}-B_{t}=0.5 B_{t}-10\) d. \(\quad B_{0}=100 \quad B_{t+1}-B_{t}=10\) e. \(\quad B_{0}=100 \quad B_{t+1}=1.2 B_{t}-5\) f. \(\quad B_{0}=100 \quad B_{t+1}-B_{t}=-0.1 B_{t}+10\) \(\begin{array}{lll}\text { g. } & B_{0}=100 & B_{t+1} & =0.9 B_{t}-10 \\\ \text { g. } & B_{0}=100 & B_{t+1} & =-0.8 B_{t}+20\end{array}\)

A one-liter flask contains one liter of distilled water and \(2 \mathrm{~g}\) of salt. Repeatedly, 50 \(\mathrm{ml}\) of solution are removed from the flask and discarded after which \(50 \mathrm{ml}\) of distilled water are added to the flask. Introduce notation and write a dynamic equation that will describe the change of salt in the beaker each cycle of removal and replacement. How much salt is in the beaker after 20 cycles of removal?

Determine the doubling times of the following exponential equations. (a) \(y=2^{t}\) (b) \(y=2^{3 t}\) (c) \(y=2^{0.1 t}\) (d) \(y=10^{t}\) (e) \(y=10^{3 t}\) (f) \(y=10^{0.1 t}\)

Two kilos of a fish poison that does not decompose are mixed into a lake that has a volume of \(100 \times 20 \times 2=4000\) cubic meters. A stream of clean water flows into the lake at a rate of 1000 cubic meters per day. Assume that it mixes immediately throughout the whole lake. Another stream flows out of the lake at a rate of 1000 cubic meters per day. a. Write a mathematical model that describes the daily change in the amount of poison in the lake. b. Let \(P_{0}, P_{1}, P_{2}, \cdots\) denote the amounts of poison in the lake, \(P_{t}\) being the amount of poison in the lake at the beginning of the \(t \underline{h}\) day after the poison is administered. Write a dynamic equation representative of the mathematical model. c. What is \(P_{0} ?\) Compute \(P_{1}\) from your dynamic equation. Compute \(P_{2}\) from your dynamic equation. d. Find a solution equation for your dynamic equation.

Equation 1.30 $$ P_{t+1}-P_{t}=b $$ represents a large number of equations $$ \begin{array}{r} P_{1}-P_{0}=b \\ P_{2}-P_{1}=b \\ P_{3}-P_{2}=b \\ \vdots \quad \vdots \quad \vdots \quad \vdots \\ P_{n-1}-P_{n-2}=b \\ P_{n}-P_{n-1}=b \end{array} $$ Add these equations to obtain $$ P_{n}=P_{0}+n b $$ Substitute \(t\) for \(n\) to obtain $$ P_{t}=P_{0}+t b $$

Show that \(A_{t}=0\) for all \(t\) is a solution to Equation 1.18 $$ A_{t+1}-A_{t}=2 \pi k \sqrt{A_{t}} $$ This means that for every \(t,\) if \(A_{t}=0\) and \(A_{t+1}\) is computed from the equation, then \(A_{t+1}=0\) (and, yes, this is simple).

Show that the doubling time of \(y=A B^{t}\) is \(1 /\left(\log _{2} B\right)\)

The equation, \(B_{t}-B_{t-1}=r B_{t-1},\) carries the same information as \(B_{t+1}-B_{t}=r B_{t}\) a. Write the first four instances of \(B_{t}-B_{t-1}=r B_{t-1}\) using \(t=1, t=2, t=3,\) and \(t=4\). b. Cascade these four equations to get an expression for \(B_{4}\) in terms of \(r\) and \(B_{0}\). c. Write solutions to and compute \(B_{40}\) for (a.) \(\quad B_{0}=50 \quad B_{t}-B_{t-1}=0.2 B_{t-1}\) (b.) \(\quad B_{0}=50 \quad B_{t}-B_{t-1}=0.1 B_{t-1}\) (c.) \(\quad B_{0}=50 \quad B_{t}-B_{t-1}=0.05 B_{t-1}\) (d.) \(\quad B_{0}=50 \quad B_{t}-B_{t-1}=-0.1 B_{t-1}\)

Two kilos of rotenone are mixed into a lake which has a volume of \(100 \times 20 \times 2=4000\) cubic meters. A stream of clean water flows into the lake at a rate of 1000 cubic meters per day. Assume that it mixes immediately throughout the whole lake. Another stream flows out of the lake at a rate of 1000 cubic meters per day. Fifteen percent of the rotenone decomposes every day. a. Write a mathematical model that describes the daily change in the amount of rotenone in the lake. b. Let \(R_{0}, R_{1}, R_{2}, \cdots\) denote the amounts of rotenone in the lake, \(R_{t}\) being the amount of rotenone in the lake at the beginning of the \(t \underline{h}\) day after the poison is administered. Write a dynamic equation representative of the mathematical model. c. What is \(R_{0}\) ? Compute \(R_{1}\) from your dynamic equation. Compute \(R_{2}\) from your dynamic equation. d. Find a solution equation for your dynamic equation.

Suppose a quail population would grow at \(20 \%\) per year without hunting pressure, and 1000 birds per year are harvested. Describe the progress of the population over 5 years if initially there are a. 5000 birds, b. 6000 birds, and c. 4000 birds.

Show that the doubling time of \(y=A 2^{k t}\) is \(1 / k\).

Suppose a population is initially of size 1,000,000 and grows at the rate of \(2 \%\) per year. What will be the size of the population after 50 years?

Observe that the graph Bacterial Growth \(\mathrm{C}\) is a plot of \(B_{t+1}-B_{t}\) vs \(B_{t}\). The points are \(\left(B_{0}, B_{1}-B_{0}\right),\left(B_{1}, B_{2}-B_{1}\right),\) etc. The second coordinate, \(B_{t+1}-B_{t}\) is the population increase during time period \(t,\) given that the population at the beginning of the time period is \(B_{t}\). Explain why the point (0,0) would be a point of this graph.

Consider a chemical reaction $$ A+B \longrightarrow A B $$ in which a chemical, \(A,\) combines with a chemical, \(B,\) to form the compound, \(A B\). Assume that the amount of \(B\) greatly exceeds the amount of \(A,\) and that in any second, the amount of \(A B\) that is formed is proportional to the amount of \(A\) present at the beginning of the second. Write a dynamic equation for this reaction, and write a solution equation to the dynamic equation.

Determine the doubling times or half-lives of the following exponential equations. (a) \(y=0.5^{t}\) (b) \(y=2^{3 t}\) (c) \(y=0.1^{0.1 t}\) (d) \(y=1000.8^{t}\) (e) \(y=45^{3 t}\) (f) \(y=0.00015^{0.1 t}\) (g) \(y=100.8^{2 t}\) (h) \(y=0.01^{3 t}\) (i) \(y=0.01^{0.1 t}\)

The polymerase chain reaction is a means of making multiple copies of a DNA segment from only a minute amounts of original DNA. The procedure consists of a sequence of, say, 30 cycles in which each segment present at the beginning of a cycle is duplicated once; at the end of the cycle that segment and one copy is present. Introduce notation and write a difference equation with initial condition from which the amount of DNA present at the end of each cycle can be computed. Suppose you begin with 1 picogram \(=0.000000000001 \mathrm{~g}\) of DNA. How many grams of DNA would be present after 30 cycles.

An egg is covered by a hen and is at \(37^{\circ} \mathrm{C}\). The hen leaves the nest and the egg is exposed to \(17^{\circ} \mathrm{C}\) air. After 20 minutes the egg is at \(34^{\circ} \mathrm{C}\). Draw a graph representative of the temperature of the egg \(t\) minutes after the hen leaves the nest. Mathematical Model. During any short time interval while the egg is uncovered, the decrease in egg temperature is proportional to the difference between the egg temperature and the air temperature. a. Introduce notation and write a dynamic equation representative of the mathematical model. b. Write a solution equation for your dynamic equation. c. Your dynamic equation should have one parameter. Use the data of the problem to estimate the parameter.

Find a formula for a population that grows exponentially and a. Has an initial population of 50 and a doubling time of 10 years. b. Has an initial population of 1000 and a doubling time of 50 years. c. Has in initial population of 1000 and a doubling time of 100 years.

Suppose \(a, b,\) and \(c\) are numbers and $$ P_{t}=a t^{2}+b t+c $$ where \(t\) is any number. Show that $$ Q_{t}=P_{t+1}-P_{t} $$ is linearly related to \(t\) (that is, \(Q_{t}=\alpha t+\beta\) for some \(\alpha\) and \(\left.\beta\right)\), and that $$ R_{t}=Q_{t+1}-Q_{t} $$ is a constant.

There is a suggestion that the world human population is growing exponentially. Shown below are the human population numbers in billions of people for the decades \(1940-2010\). \(\begin{array}{lrrrrrrrr}\text { Year } & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\ \text { Index, } t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \text { Human Population } \times 10^{6} & 2.30 & 2.52 & 3.02 & 3.70 & 4.45 & 5.30 & 6.06 & 6.80\end{array}\) 1\. Test the equation $$ P_{t}=2.21 .19^{t} $$ against the data where \(t\) is the time index in decades after 1940 and \(P_{t}\) is the human population in billions. 2\. What percentage increase in human population each decade does the model for the equation assume? 3\. What world human population does the equation predict for the year \(2050 ?\)

What initial condition and dynamic equation would describe the growth of an Escherichia coli population in a nutrient medium that had 250,000 E. coli cells per milliliter at the start of an experiment and one-fourth of the cells divided every 30 minutes.

The nitrogen partial pressure in a muscle of a scuba diver is initially 0.8 atm. She descends to 30 meters and immediately the \(\mathrm{N}_{2}\) partial pressure in her blood is \(2.4 \mathrm{~atm},\) and remains at 2.4 atm while she remains at 30 meters. Each minute the \(\mathrm{N}_{2}\) partial pressure in her muscle increases by an amount that is proportional to the difference in 2.4 and the partial pressure of nitrogen in her muscle at the beginning of that minute. a. Write a dynamic equation with initial condition to describe the \(\mathrm{N}_{2}\) partial pressure in her muscle. b. Your dynamic equation should have a proportionality constant. Assume that constant to be 0.067. Write a solution to your dynamic equation. c. At what time will the \(\mathrm{N}_{2}\) partial pressure be \(1.6 ?\) d. What is the half-life of \(N_{2}\) partial pressure in the muscle, with the value of \(K=0.067 ?\)

Light intensities, \(I_{1}\) and \(I_{2}\), are measured at depths \(d\) in meters in two lakes on two different days and found to be approximately $$ I_{1}=22^{-0.1 d} \quad \text { and } \quad I_{2}=42^{-0.2 d} $$ a. What is the half-life of \(I_{1} ?\) b. What is the half-life of \(I_{2} ?\) c. Find a depth at which the two light intensities are the same. d. Which of the two lakes is the muddiest?

a. The mass of a single \(V\). natriegens bacterial cell is approximately \(210^{-11}\) grams. If at time 0 there are \(10^{8} \mathrm{~V}\). natriegens cells in your culture, what is the mass of bacteria in your culture at time \(0 ?\) b. We found the doubling time for \(V\). natriegen to be 22 minutes. Assume for simplicity that the doubling time is 30 minutes and that the bacteria continue to divide at the same rate. How may minutes will it take to have a mass of bacteria from Part a. equal one gram? c. The earth weighs \(610^{27}\) grams. How many minutes will it take to have a mass of bacteria equal to the mass of the Earth? How many hours is this? Why aren't we worried about this in the laboratory? Why hasn't this happened already in nature? Explain why it is not a good idea to extrapolate results far beyond the end-point of data gathering.

Show that \(y=A B^{t}\) with \(B<1\) has a half-life of $$ T_{\text {Half }}=\frac{\log \frac{1}{2}}{\log B}=\frac{-\log 2}{\log B} $$