Growth of Exponential Functions
at + Infinity

by
K. D.  Stroyan
University of Iowa

copyright 1997 by Academic Press, Inc. - All rights reserved.

Mathematica Functions

Needs["Graphics`Colors`"];

Notebook Overview

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.

\(Plot[2\^\(t/6\), {t, 0, 24}, PlotStyle -> {Blue}, 
      PlotLabel -> One\ Day, AxesLabel -> {"hours", "slime"}]; 

    Plot[2\^\(t/6\), {t, 0, 24*30}, PlotStyle -> {Red}, 
      PlotLabel -> One\ Month, AxesLabel -> {"hours", "slime"}]; \)

References to the Text

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.

Magnitudes in the Algae Growth Model

Consider the algae growth model  N(t) = No  (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

(2^(t/6)) tends to  ∞

Our algae growth function is defined in the next cell.  Open and Enter it now.

The Algae Growth Model

The next cell enters n as a function of  t.  (Notice the underscore on the left side.  This tells Mathematica that  t is the input.)  Enter it now.

\(Clear[n, t]; 
n[t_] := n0\ 2\^\(t/6\); 
n[t]\)

Check on how many cells there are in  6  hours by <Enter>ing the next cell.  How many are there supposed to be?

n[6]

n[3]

Use the next empty input cell to compute  n[3]  yourself.  Get both symbolic and numerical answers.

n[3]

N[n[3]]

n[ one week ]

How many algae cells are there after a whole week of this prolific reproduction?  Enter the next computation to see.

\(t\  = \ \(?? ?\ ***Number\ of\ Hours\ in\ One\ Week***
n[t]\)\)

N[n[t]]

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?

n[ one month]

Use 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?

\(t\  = \ 
      \(?? ?\ ***Number\ of\ Hours\ in\ Thirty\ Days***
N[n[t]]\)\)

n[t]

Approximate Mass of Lake Michigan

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/cm3, the value of lm will be the mass of all the water in Lake Michigan in grams (roughly.)

lm = 
      (80 (10^3) (10^2)) (320 (10^3) (10^2)) (30 (10^2))

N[lm]

Mass of Algae vs. Mass of Lake Michigan

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.)

t = 7*24N[n[t]/lm]

n[ 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?

t = 30*24N[n[t]/lm]

Filling Lake Michigan

How long would it take 1000  1 mg  algae cells doubling every  6  hours to fill Lake Michigan completely?

Clear[t]     \(n0\  = \ 1	\ \ \  (*\ 
      This\ makes\ \ n\ \ the\ mass\ as\ a\ function\ of\ \ \(t . \)\ *) 

    \)Solve[ n[t]  ==  lm ,  t]

N[%]  (*This gives the numerical value of the previous output.*)

N[357/24]   (* Answer in days. *)

Problem 28.2

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?