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Cell[CellGroupData[{
Cell[TextData[StyleBox["Growth of Exponential Functions\nat + Infinity",
FontSize->18]], "Title",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
by
K. D. Stroyan
University of Iowa\
\>", "Subsubtitle",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
copyright 1997 by Academic Press, Inc. - All rights reserved.\
\>", "Text",
Evaluatable->False,
AspectRatioFixed->True,
FontFamily->"Times"],
Cell[CellGroupData[{
Cell[TextData[{
StyleBox["Mathematica",
FontSlant->"Italic"],
" Functions"
}], "Subsubsection"],
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\(\(Needs["\"]; \)\)], "Input",
InitializationCell->True]
}, Closed]],
Cell[CellGroupData[{
Cell["Notebook Overview ", "Section",
Evaluatable->False,
AspectRatioFixed->True],
Cell["\<\
Exponential functions tend to infinity very rapidly. This NoteBook shows you \
just how rapidly algae that double every 6 hours would grow. This growth \
is so rapid that real algae can not continue to double at their natural \
fertility rate - they run out of food or space. Enter the plot commands in \
the next cell showing one day's growth and one month's.\
\>", "Text",
Evaluatable->False],
Cell[BoxData[
\(Plot[2\^\(t/6\), {t, 0, 24}, PlotStyle -> {Blue},
PlotLabel -> One\ Day, AxesLabel -> {"\", "\"}]; \n
Plot[2\^\(t/6\), {t, 0, 24*30}, PlotStyle -> {Red},
PlotLabel -> One\ Month, AxesLabel -> {"\", "\"}]; \)],
"Input"],
Cell[CellGroupData[{
Cell["References to the Text", "Subsection",
Evaluatable->False],
Cell["\<\
This NoteBook goes with Problem 28.2 on High School Review of the text. It \
is closely related to Section 8.3 on growth of logs, exponentials, and \
powers.\
\>", "Text",
Evaluatable->False]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["Magnitudes in the Algae Growth Model", "Section",
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Cell[TextData[{
"Consider the algae growth model N(t) = N",
StyleBox["o",
FontSize->11,
FontVariations->{"CompatibilityType"->"Subscript"}],
" ",
Cell[BoxData[
\(2\^\(t/6\)\)]],
" from Probelm 28.2. This model simply says the number of cells doubles \
every 6 hours. How long could this go on? We want you to develop an \
intuitive feel for how fast"
}], "Text",
Evaluatable->False],
Cell[TextData[{
"\t",
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Cell["\<\
Our algae growth function is defined in the next cell. Open and Enter it \
now.\
\>", "Text",
Evaluatable->False],
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Evaluatable->False],
Cell[TextData[{
"The next cell enters n as a function of t. (Notice the underscore on the \
left side. This tells ",
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" that t is the input.) Enter it now."
}], "Text"],
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Cell["\<\
Check on how many cells there are in 6 hours by ing the next cell. \
How many are there supposed to be?\
\>", "Text",
Evaluatable->False],
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\(n[6]\)], "Input"]
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Cell[CellGroupData[{
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Use the next empty input cell to compute n[3] yourself. Get both symbolic \
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\>", "Text",
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Cell["N[n[3]]", "Input"]
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How many algae cells are there after a whole week of this prolific \
reproduction? Enter the next computation to see.\
\>", "Text",
Evaluatable->False],
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\(t\ = \ \(?? ?\ ***Number\ of\ Hours\ in\ One\ Week***\nn[t]\)\)],
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Cell["\<\
Hummm, ten to the 8th power, huh? Let's say one thousand algae cells weigh \
one gram and we started with only one gram. That means the weight of the \
algae is 2.68435 x 10^8 grams after a week. How much is that in intuitive \
terms?\
\>", "Text",
Evaluatable->False]
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Cell["n[ one month]", "Subsubsection",
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Cell["\<\
\tUse the next empty input cell to compute the number of algae cells in one \
month (assuming that the doubling continues every 6 hours.) How much is \
the mass of all the algae if we start with 1,000 cells each weighing 1 mg?\
\
\>", "Text",
Evaluatable->False],
Cell[BoxData[
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\(?? ?\ ***Number\ of\ Hours\ in\ Thirty\ Days***\nN[n[t]]\)\)], "Input"],
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Cell["Approximate Mass of Lake Michigan", "Subsubsection",
Evaluatable->False],
Cell[TextData[{
"Lake Michigan is about 50 miles wide and 200 miles long. Let's say it is \
100 feet deep. How much does all the water in Lake Michigan weigh? It might \
help to convert to all metric units. Let's say Lake Michigan is 80 km wide, \
320 km long and 30 m deep. The next cell multiplies each dimension of the \
lake in centimeters. That gives a volume in cubic centimeters. Since the \
density of fresh water is 1 gram/cm",
StyleBox["3",
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\((80\ 10\^3\ 10\^2)\)\ \((320\ 10\^3\ 10\^2)\)\ \((30\ 10\^2)\)\ (*
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n[ one week ] is pretty big. How much of Lake Michigan would be filled - \
packed tight with no water? In other words, how much is Mass[n[ one week ]] \
/ Mass[ Lake Michigan ]? (Compute this in the next cell.)\
\>", "Text",
Evaluatable->False],
Cell[BoxData[{
\(t = 7*24\),
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Cell["\<\
\tn[ three months ] is BIG. How many copies of Lake Michigan would be filled \
- packed tight with no water? (Compute this in the next cell.) \
Approximately how big is the Pacific Ocean?\
\>", "Text",
Evaluatable->False],
Cell[BoxData[{
\(t = 30*24\),
\(N[n[t]\/lm]\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
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Cell["\<\
\tHow long would it take 1000 1 mg algae cells doubling every 6 hours to \
fill Lake Michigan completely?\
\>", "Text",
Evaluatable->False],
Cell[BoxData[{
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This\ makes\ \ t\ \ a\ symbolic\ variable\ \(again . \)\ *) \),
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Cell["\<\
N[%] (*This gives the numerical value of the previous output.*)\
\>", "Input"],
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Cell[CellGroupData[{
Cell["Problem 28.2", "Section",
Evaluatable->False,
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Cell["\<\
Algae cells (with sufficient space and food) double every 6 hours. How \
many cells are there after three hours? (Refer to the answer in n[3] you can \
obtain the answer.) How many cells are there in one day? How many in one \
week? If we start with 1000 cells weighing a total of 1 gram, what is \
the weight of the algae at the end of 1 month? How does this mass compare \
with the mass of a typical little pond like Lake Michigan?\
\>", "Text",
Evaluatable->False]
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}, Open ]]
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