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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 385307, 8447]*) (*NotebookOutlinePosition[ 385986, 8470]*) (* CellTagsIndexPosition[ 385942, 8466]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Mathematics Linear, Geometric and Exponential Growth\ \>", "Subtitle", TextAlignment->Center], Cell[TextData[{ "\tHow fast things grow (and the opposite, how fast things decay) is an \ important issue for scientists in many areas. With this lesson we will use ", StyleBox["Mathematica", FontSlant->"Italic"], " to explore the differences between Linear, Geometric and Exponential \ growth.\n\tThe Scenario: A long-lost, wealthy aunt has recently passed away, \ leaving you as one of three beneficiaries in her will. You are given the \ choice between three stock investments, which pay the following returns over \ the next year-and-a-half.\n\tLines & Things: $10,000 per month for 18 months.\ \n\tAmerican Explus : $1,500, first month. Add $1,500 to the previous payment \ for each following month ($3,000 the second, $4,500 the third, $6,000 the \ fourth, and so on) for 18 months\n\tRatio Shack:: $1 (yes, one dollar), first \ month. Multiply the previous payment by two for each following month ($2 the \ second, $4 the third, $8, $16 and so on) for 18 months.\n\tThe stocks you do \ ", StyleBox["not", FontWeight->"Bold"], " choose will be used to maintain a bountiful supply of Luv Your Kitty \ gourmet cat food for your aunt's great white persian, Cujette, and to \ maintain the family mansion for the her beloved komodo dragon, Fluffles. \ Which choice is best for ", StyleBox["your", FontWeight->"Bold"], " pocketbook? " }], "Text", FontSize->14], Cell[TextData[{ "\t", StyleBox["Mathematica", FontSlant->"Italic"], " is an excellent calculator for a problem like this, especially because we \ can keep track of results as we go along ", StyleBox["and", FontWeight->"Bold"], " use plotting functions to help visualize the mathematical relationships. \ Lets begin by defining the number of months for payment." }], "Text", FontSize->14], Cell[BoxData[ \(\(NumMonths\ = \ 18;\)\)], "Input"], Cell[TextData[{ "\tStarting with the first option, we define a variable to keep track of \ the total amount paid out (", StyleBox["LTtotal", "Input"], "), which at first is zero." }], "Text", FontSize->14], Cell[BoxData[ \(\(LTtotal\ = \ 0;\)\)], "Input"], Cell["\<\ \tWe will also use a variable to keep track of how much is paid out \ each month, which in the case of Lines & Things, is a constant value of \ $10,000.\ \>", "Text", FontSize->14], Cell[BoxData[ \(\(LTmonthly\ = \ 10000;\)\)], "Input"], Cell["\<\ \tTo figure out how much we have made after the first month, we \ simply add the current monthly income to the current total.\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(LTtotal\ + \ LTmonthly\)], "Input"], Cell[BoxData[ \(10000\)], "Output"] }, Open ]], Cell[TextData[{ "\tIn order to keep track of this value, however, ", StyleBox["Mathematica", FontSlant->"Italic"], " needs to assign it to a variable. We assign the value to ", StyleBox["LTtotal", "Input"], ", because it is, after all, the variable we are using to keep track of our \ total!" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(LTtotal\ = \ LTtotal\ + \ LTmonthly\)], "Input"], Cell[BoxData[ \(10000\)], "Output"] }, Open ]], Cell["\<\ \tOur total after one month is $10,000. Executing the same command \ gives us the total after two months:\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(LTtotal\ = \ LTtotal\ + \ LTmonthly\)], "Input"], Cell[BoxData[ \(20000\)], "Output"] }, Open ]], Cell[TextData[{ "\tThere is a short-cut syntax for adding a value to a running total. The \ following command adds ", StyleBox["LTmonthly", "Input"], " to the current ", StyleBox["LTtotal", "Input"], " to get the 'next' value, in this case for month three." }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(LTtotal\ += \ LTmonthly\)], "Input"], Cell[BoxData[ \(30000\)], "Output"] }, Open ]], Cell[TextData[{ "\t", StyleBox["We want to repeat this process some number of times equal to ", FontSize->14], StyleBox["NumMonths", "Input", FontSize->14], StyleBox[" (in this case a total of 18 times). We let ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[" do the work for us by using a ", FontSize->14], StyleBox["loop ", FontSize->14, FontSlant->"Italic"], StyleBox["function. One common loop function takes the form ", FontSize->14], StyleBox["Do[", "Input", FontSize->14], StyleBox["expr,", "Input", FontSize->14, FontSlant->"Italic"], StyleBox["{i,", "Input", FontSize->14], StyleBox["imin,imax", "Input", FontSize->14, FontSlant->"Italic"], StyleBox["}]", "Input", FontSize->14], StyleBox[". The loop uses a ", FontSize->14], StyleBox["counter", FontSize->14, FontSlant->"Italic"], StyleBox[", in this case represented by the variable ", FontSize->14], StyleBox["i", "Input", FontSize->14], StyleBox[", to count from ", FontSize->14], StyleBox["imin", "Input", FontSize->14, FontSlant->"Italic"], StyleBox[" to ", FontSize->14], StyleBox["imax", "Input", FontSize->14, FontSlant->"Italic"], StyleBox[", each time evaluating ", FontSize->14], StyleBox["expr", "Input", FontSize->14, FontSlant->"Italic"], StyleBox[". To repeat our addition for the right number of months, we \ simply place the command inside a ", FontSize->14], StyleBox["Do", "Input", FontSize->14], StyleBox[" loop. Notice that we reset ", FontSize->14], StyleBox["LTtotal", "Input", FontSize->14], StyleBox[" to zero before doing the loop (Why?)", FontSize->14] }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(\(LTmonthly\ = \ 10000;\)\), "\[IndentingNewLine]", \(\(LTtotal\ = \ 0;\)\), "\[IndentingNewLine]", \(Do[\[IndentingNewLine]LTtotal\ += \ LTmonthly, \[IndentingNewLine]{i, 1, NumMonths}]\), "\[IndentingNewLine]", \(LTtotal\)}], "Input"], Cell[BoxData[ \(180000\)], "Output"] }, Open ]], Cell[TextData[{ "\tAs expected, the total payout is $180,000 (18 months times $10,000). \ However, what if we want to know intermediate amounts, like how much we have \ after one year (12 months). We could redo the calculation, changing ", StyleBox["NumMonths", "Input"], " to twelve, however, that could get tedious if we need to do it for each \ month. Notice, however, that during the calculation, ", StyleBox["Mathematica", FontSlant->"Italic"], " actually did calculate the total for each month (it was stored in ", StyleBox["LTtotal", "Input"], "). One way to keep track of that information is to use the ", StyleBox["Table", "Input"], " command, which acts like the ", StyleBox["Do", "Input"], " command as far as repeating an expression, but actually stores the \ results in a list, which in this case we will name ", StyleBox["LTList", "Input"], StyleBox[".", "Subsubsection"], " " }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(\(LTmonthly\ = \ 10000;\)\), "\[IndentingNewLine]", \(\(LTtotal\ = \ 0;\)\), "\[IndentingNewLine]", \(LTList\ = \ Table[\[IndentingNewLine]LTtotal\ += \ LTmonthly, \[IndentingNewLine]{i, 1, NumMonths}]\)}], "Input"], Cell[BoxData[ \({10000, 20000, 30000, 40000, 50000, 60000, 70000, 80000, 90000, 100000, 110000, 120000, 130000, 140000, 150000, 160000, 170000, 180000}\)], "Output"] }, Open ]], Cell["\<\ \tNow if we want to know what the total is after a particular \ month, we can simply look at a certain part of the list. 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00Goool002Ooool00`000?ooooooo`3foooo000Woooo00<0003oooooool0m_ooo`009oooo`030000 oooooooo0?Koool002Ooool00`000?ooooooo`3foooo003ooooo8Oooo`00ooooob7oool00?oooolQ oooo003ooooo8Oooo`00ooooob7oool00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-2.91646, -21650.6, \ 0.0744479, 1204.59}}] }, Open ]], Cell["\<\ \tNow let's take a look at option #2, \"American Explus\". We can \ begin by setting the total and monthly payouts\ \>", "Text", FontSize->14], Cell[BoxData[{ \(\(AEtotal\ = \ 0;\)\), "\[IndentingNewLine]", \(\(AEmonthly\ = \ 1500;\)\)}], "Input"], Cell[TextData[{ "\tNotice, however, that each month the total changes, ", StyleBox["and", FontWeight->"Bold"], " the amount per month changes! Each month we need to add $1500 to the \ monthly payout, as well as add the payout to the total. " }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(\(AEmonthly\ += \ 1500;\)\), "\[IndentingNewLine]", \(AEtotal\ += \ AEmonthly\)}], "Input"], Cell[BoxData[ \(3000\)], "Output"] }, Open ]], Cell[TextData[{ "\tWait a minute! After month one we should only have $1500! What \ happened?\n\tNotice that we added $1500 dollars to ", StyleBox["AEmonthly", "Input"], " ", StyleBox["before", FontSlant->"Italic"], " adding ", StyleBox["AEmonthly", "Input"], " to ", StyleBox["AEtotal", "Input"], ", which means our first monthly payment was $3000 and not $1500. There \ are two ways we can fix this - either switch the order of the additions, or \ start ", StyleBox["AEmonthly", "Input"], " at zero. Either way we need to reset variables and start over. Here we \ decide to keep the same order, but start the monthly payments at zero. This \ means that after every calculation, ", StyleBox["AEmonthly", "Input"], " will be storing the ", StyleBox["most recent monthly payment", FontSlant->"Italic"], ". If we choose the other method, ", StyleBox["AEmonthly", "Input"], " would instead store the ", StyleBox["next expected payment.", FontSlant->"Italic"], " Notice that we must be careful and pay attention to what is really stored \ in each variable! \n\tWe would still like to set our variables at the \ beginning, though, so we add a variable ", StyleBox["AEaddmonthly", "Input"], " for the regular monthly ", StyleBox["increase ", FontSlant->"Italic"], "in payments." }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(\(AEaddmonthly\ = \ 1500;\)\), "\[IndentingNewLine]", \(\(AEmonthly\ = \ 0;\)\), "\[IndentingNewLine]", \(\(AEtotal\ = \ 0;\)\), "\[IndentingNewLine]", \(\(AEmonthly\ += \ AEaddmonthly;\)\), "\[IndentingNewLine]", \(AEtotal\ += \ AEmonthly\)}], "Input"], Cell[BoxData[ \(1500\)], "Output"] }, Open ]], Cell[TextData[{ "\tWe again use the ", StyleBox["Table", "Input"], " function to generate a list of results. Notice that because we must to \ more than one calculation inside the table, we use parentheses to group them. \ We are also potentially interested in how the monthly salary changes, i.e. \ what the value of ", StyleBox["AEmonthly", "Input"], " is, so we store ", StyleBox["both ", FontSlant->"Italic"], "values as a short list in the table." }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(\(AEaddmonthly\ = \ 1500;\)\), "\[IndentingNewLine]", \(\(AEmonthly\ = \ 0;\)\), "\[IndentingNewLine]", \(\(AEtotal\ = \ 0;\)\), "\[IndentingNewLine]", \(AEList\ = \ Table[\[IndentingNewLine]\((AEmonthly\ += \ AEaddmonthly; \[IndentingNewLine]AEtotal\ += \ AEmonthly; \[IndentingNewLine]{AEmonthly, AEtotal})\), \[IndentingNewLine]{i, 1, NumMonths}]\)}], "Input"], Cell[BoxData[ \({{1500, 1500}, {3000, 4500}, {4500, 9000}, {6000, 15000}, {7500, 22500}, {9000, 31500}, {10500, 42000}, {12000, 54000}, {13500, 67500}, {15000, 82500}, {16500, 99000}, {18000, 117000}, {19500, 136500}, {21000, 157500}, {22500, 180000}, {24000, 204000}, {25500, 229500}, {27000, 256500}}\)], "Output"] }, Open ]], Cell["\<\ \tFor any given month, the first number represents the monthly \ payment for that month, and the second number represents the total payout up \ to that point.\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(AEList[\([8]\)]\)], "Input"], Cell[BoxData[ \({12000, 54000}\)], "Output"] }, Open ]], Cell["\<\ \tWe can access a monthly value by itself by using an index for the \ first part .\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(AEList[\([8, 1]\)]\)], "Input"], Cell[BoxData[ \(12000\)], "Output"] }, Open ]], Cell["\<\ \tSimilarly, we can access a monthly total with an index for the \ second part .\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(AEList[\([8, 2]\)]\)], "Input"], Cell[BoxData[ \(54000\)], "Output"] }, Open ]], Cell[TextData[{ "\tIt is also a simple matter to show ", StyleBox["All", "Input"], " the monthly totals:" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(AEList[\([All, 2]\)]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \({1500, 4500, 9000, 15000, 22500, 31500, 42000, 54000, 67500, 82500, 99000, 117000, 136500, 157500, 180000, 204000, 229500, 256500}\)], "Output"] }, Open ]], Cell[TextData[{ "\tOnce again we can use ", StyleBox["ListPlot", "Input"], " to see the progression of earnings." }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\(AEtotgraph = ListPlot[AEList[\([All, 2]\)], PlotJoined \[Rule] True, PlotStyle \[Rule] {RGBColor[0, 1, 0]}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0529101 0.0147151 2.29475e-006 [ 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", StyleBox["Geometric", FontWeight->"Bold"], " growth is actually characterized by a linear rate of change.\n\tOn to the \ third option! We should be able to copy the process from option two, with \ some minor changes..." }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(\(RSmultmonthly\ = \ 2;\)\), "\[IndentingNewLine]", \(\(RSmonthly\ = \ 1.0;\)\), "\[IndentingNewLine]", \(\(RStotal\ = \ 0;\)\), "\[IndentingNewLine]", \(RSList\ = \ Table[\[IndentingNewLine]\((RSmonthly\ *= \ RSmultmonthly; \[IndentingNewLine]RStotal\ += \ RSmonthly; \[IndentingNewLine]{RSmonthly, RStotal})\), \[IndentingNewLine]{i, 1, NumMonths}]\)}], "Input"], Cell[BoxData[ \({{2.`, 2.`}, {4.`, 6.`}, {8.`, 14.`}, {16.`, 30.`}, {32.`, 62.`}, {64.`, 126.`}, {128.`, 254.`}, {256.`, 510.`}, {512.`, 1022.`}, {1024.`, 2046.`}, {2048.`, 4094.`}, {4096.`, 8190.`}, {8192.`, 16382.`}, {16384.`, 32766.`}, {32768.`, 65534.`}, {65536.`, 131070.`}, {131072.`, 262142.`}, {262144.`, 524286.`}}\)], "Output"] }, Open ]], Cell[TextData[{ "\tAre the results correct? Apparently not, as in the first month we made \ $2, not the expected $1. Again the problem comes up with what exactly is \ being stored. Before we start the simulation, we need to set a value for ", StyleBox["RSmonthly", "Input"], ". However, the first thing that happens is that the value is multiplied by \ ", StyleBox["RSmultmonthly", "Input"], ". In order for the first month to be correct, we need to divide the \ expected value (1.0) by the multiplier (in this case, 2). Let's look at the \ revised version:" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[{ \(\(RSmultmonthly\ = \ 2;\)\), "\[IndentingNewLine]", \(\(RSmonthly\ = \ 1.0/RSmultmonthly;\)\), "\[IndentingNewLine]", \(\(RStotal\ = \ 0;\)\), "\[IndentingNewLine]", \(RSList\ = \ Table[\[IndentingNewLine]\((RSmonthly\ *= \ RSmultmonthly; \[IndentingNewLine]RStotal\ += \ RSmonthly; \[IndentingNewLine]{RSmonthly, RStotal})\), \[IndentingNewLine]{i, 1, NumMonths}]\)}], "Input"], Cell[BoxData[ \({{1.`, 1.`}, {2.`, 3.`}, {4.`, 7.`}, {8.`, 15.`}, {16.`, 31.`}, {32.`, 63.`}, {64.`, 127.`}, {128.`, 255.`}, {256.`, 511.`}, {512.`, 1023.`}, {1024.`, 2047.`}, {2048.`, 4095.`}, {4096.`, 8191.`}, {8192.`, 16383.`}, {16384.`, 32767.`}, {32768.`, 65535.`}, {65536.`, 131071.`}, {131072.`, 262143.`}}\)], "Output"] }, Open ]], Cell["\<\ \tThat looks about right, for the first few months anyway. 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As is true of any ", StyleBox["exponential", FontSlant->"Italic"], " growth function, the growth itself, as well as its rate of change, will \ demonstrate a characteristic period of slow increase followed by increasingly \ rapid increases over time", ".\n\tLet us compare the final totals for each option from the total values \ after month 18.\t" }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \({LTList[\([NumMonths]\)], AEList[\([NumMonths, 2]\)], RSList[\([NumMonths, 2]\)]}\)], "Input"], Cell[BoxData[ \({180000, 256500, 262143.`}\)], "Output"] }, Open ]], Cell["\<\ \tApparently the Ratio Shack stock, in spite of the $1 starting \ payout, is the most valuable after 18 months. Would this still be true if \ there were only 12 months of payouts?\ \>", "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \({LTList[\([12]\)], AEList[\([12, 2]\)], RSList[\([12, 2]\)]}\)], "Input"], Cell[BoxData[ \({120000, 117000, 4095.`}\)], "Output"] }, Open ]], Cell[TextData[{ "\tApparently not! In fact, the Ratio Shack total after 12 months is quite \ a bit lower than the other two, and the regular payments from Lines & Things \ make the most sense if the limit is twelve months. \n\tWe've given names to \ each of our plots, so it is now a simple matter to ", StyleBox["Show", "Input"], " them all simultaneously, which can give us a better picture of the \ relative worth of the payouts.We use ", StyleBox["Table", "Input"], " to generate lists of locations and specify ", StyleBox["Ticks", "Input"], " on the horizontal axes and ", StyleBox["Gridlines", "Input"], " for in both directions." }], "Text", FontSize->14], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[LTtotgraph, AEtotgraph, RStotgraph, \[IndentingNewLine]Ticks\ \[Rule] \ {Table[i, {i, 18}], Automatic}, GridLines \[Rule] {Table[i, {i, 18}], Table[50000\ i, {i, 5}]}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix 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When \ does that change? Which months are best using American Explus? When does ", StyleBox["that", FontWeight->"Bold"], " change?" }], "Text", FontSize->14], Cell[TextData[{ "\tNow for the final challenge...\n\tWith the newfound wealth from your \ aunt, you have purchased one hundred acres of prime farmland in North \ Carolina. You have three options as to how to use the land.\n\tOption 1: \ Plant strawberries! You estimate that profits from 'self-pick' sales should \ be about $1,400 ", StyleBox["per acre", FontSlant->"Italic"], " per year.\n\tOption 2: Raise hogs! After investing your remaining money \ in hog pens and hog-waste treatment facilities, you should be able to \ purchase ten breeding sows to start your operation. You figure that for ", StyleBox["each", FontSlant->"Italic"], " sow you can count on an annual brood of five 'prime' piglets to be \ fattened and sold to market at $1,000 per head. You also plan to increase \ your number of breeding sows by two each year through minor revenue from \ fertilizer sales. \n\tOption 3: Raise chickens! After covering the land with \ chicken pens, you should have just enough money for two laying hens, each of \ which will provide $5 per year profit in egg sales. Adding a rooster or two \ as well as a chick nursery, you figure that you should be able to add 50% to \ your number of laying hens every year as well." }], "Text", FontSize->14], Cell["\<\ \tHow long do you want to stay in farming? After all the effort of \ buying the farm, you figure at least five years, but probably less than \ thirty-five. Pick a number of years to work on the farm. Which option do you \ suppose will earn you the most money in that period of time? \tModify the functions from the first example to calculate and plot the total \ profits for each farming option. For what years is option 1 the best? Option \ 2? Option 3? \tHow long would you have two work to make $1 million with each type of farm? \ How long to make $5 million? \ \>", "Text", FontSize->14], Cell[TextData[{ "\tWhenever we find ourselves repeating a calculation or a process, it \ often makes sense to define a reusable program or function in ", StyleBox["Mathematica", FontSlant->"Italic"], " or another language to make things easier. If we notice that all of our \ growth calculations have some similar elements, we can design a function that \ includes those similar pieces, and allows us to change the other pieces.\n\t\ In the function ", StyleBox["GrowthFunc", "Input"], ", we expect four ", StyleBox["parameters", FontSlant->"Italic"], ", i.e. pieces of information that can change depending on how the function \ is used. ", StyleBox["startval", "Input"], " is the expected value for the first time period, ", StyleBox["multip", "Input"], " is the amount the regular 'payout' will be multiplied by each time \ period, ", StyleBox["addin", "Input"], " is the amount added to the regular growth for each time period, and ", StyleBox["length", "Input"], " is the number of time periods, as well as the length of the list. Notice \ that multiplication occurs before addition, and that in order to get the \ first month correct, we subtract and then divide the inital value at the very \ beginning.", Cell[BoxData[ FormBox[Cell[""], TraditionalForm]]], "\n\t", " The ", StyleBox[":=", "Input"], " assignment is used to tell ", StyleBox["Mathematica", FontSlant->"Italic"], " \"don't evaluate this now, evaluate it when it is called for later\", and \ is the correct form for defining a function." }], "Text", FontSize->14], Cell[BoxData[ \(GrowthFunc[startval_, multip_, addin_, length_]\ := \[IndentingNewLine]\((GFmonthly\ = \ \((startval - addin)\)/multip; \[IndentingNewLine]GFtotal\ = \ 0; \[IndentingNewLine]Table[\[IndentingNewLine]\((GFmonthly\ *= \ multip; \[IndentingNewLine]GFmonthly\ += \ addin; \[IndentingNewLine]GFtotal\ += \ GFmonthly; \[IndentingNewLine]{GFmonthly, GFtotal})\), \[IndentingNewLine]{i, 1, length}])\)\)], "Input"], Cell[TextData[{ "\tSo how do we calculate for each option? In the first case, the starting \ value is $1400 for each of 100 acres, so we put in ", StyleBox["1400*100", "Input"], " in the first location, which represents the ", StyleBox["startval", "Input"], ". We don't want the annual value to increase, so we use a multiplier \ value of ", StyleBox["1", "Input"], " and an add-in value of ", StyleBox["0", "Input"], ", which will leave the value as-is. The final value indicates that we are \ interested in calculations for a span of ", StyleBox["32", "Input"], " years. ", " " }], "Text", FontSize->14], Cell[BoxData[ \(\(Opone\ = \ GrowthFunc[1400*100, 1, 0, 32];\)\)], "Input"], Cell[TextData[{ "\tFor the second option, we expect $1000 for each of 5 offspring from each \ of 10 sows for a ", StyleBox["startval", "Input"], " of ", StyleBox["1000*5*10", "Input"], ". Annual value will increase linearly, so we still use a multiplier value \ of ", StyleBox["1", "Input"], ", but expect a regular increase of $1000 for each of 5 offspring for 2 new \ sows every year (", StyleBox["1000*5*2", "Input"], "), again for ", StyleBox["32", "Input"], " years. ", " " }], "Text", FontSize->14], Cell[BoxData[ \(\(Optwo\ = \ GrowthFunc[1000*5*10, 1, 1000*5*2, 32];\)\)], "Input"], Cell[TextData[{ "\tThe third option starts with a small payout of only $5 for each of two \ hens (", StyleBox["5*2", "Input"], "), but has a multiplier value of ", StyleBox["1.5", "Input"], ",which we might well expect to make up for no linear increase (add-in \ value of ", StyleBox["0", "Input"], "), epecially over the course of ", StyleBox["32", "Input"], " years. ", " " }], "Text", FontSize->14], Cell[BoxData[ \(\(Opthree\ = \ GrowthFunc[5*2, 1.5, 0, 32];\)\)], "Input"], Cell[TextData[{ "\tNow we can plot the values of the three options and ", StyleBox["Show", "Input"], " them in a single graph for comparison. As might be expected, the results \ are similar to those for example 1. What might you change to make the graphs \ look different? 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