(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 114001, 2536]*) (*NotebookOutlinePosition[ 115056, 2572]*) (* CellTagsIndexPosition[ 115012, 2568]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Name: Roberta Jones"], "Section", Editable->False, Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->False], Cell[TextData["Project: Strategies for solving problems."], "Section", Editable->False, Evaluatable->False, PageBreakAbove->False, AspectRatioFixed->False], Cell[TextData[ "Description: This lesson models several different ways of solving an \ application problem. The problem is one typically found in an Algebra I \ text. These include using a table, graphing with two variables, graphing \ with one variable, and solving algebraically. Examples are given of how to \ extend the problem in several interesting ways."], "Section", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[TextData[{ StyleBox[" \n \n ", Editable->False, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->False], StyleBox[" Problem Solving Using \n Several Techniques ", Editable->False, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->False, FontColor->RGBColor[0.533333, 0.0576181, 0.213443]], StyleBox[" ", Editable->False, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->False] }], "Title", Editable->False, Evaluatable->False, PageBreakAbove->True, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData[" Here's a Problem"], "Section", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData[ "A farmer with 2986 meters of fencing wants to enclose a rectangular field \ that borders on a straight highway which has an existing fence. So the \ farmer needs to use his fence for three sides of the field. He also wants \ the area of the field to be 832,000 square meters. Find the dimensions of \ the field to the nearest whole number.\n\nSome people may think there is one \ and only one way to solve this problem. That is not the case. The purpose \ of this exercise is to point out different ways of solving a problem. Please \ keep that in mind as you read the following pages.\n\nFirst, let's draw a \ picture to see what the situation is."], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart % Scaling calculations 0.2 0.05 0.24721 0.0618 [ [ 0 0 0 0 ] [ 1 0.618034 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave grestore grestore 0 0 moveto 1 0 lineto 1 0.618034 lineto 0 0.618034 lineto closepath clip newpath 0 setgray gsave gsave 0 setgray [(Highway)] 0.1535 0.37082 0 0 Mshowa grestore gsave 0 setgray [(? = length)] 0.5 0.43262 0 0 Mshowa grestore gsave 0 setgray [(width = x)] 0.3 0.18541 0 0 Mshowa grestore gsave 0.85 setgray 0.25 0.24721 moveto 0.25 0.86525 lineto 0.4 0.86525 lineto 0.4 0.24721 lineto fill grestore grestore % End of Graphics MathPictureEnd\ \>"], "Graphics", Editable->False, 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In the first column, put the width. In the \ second column put the other width. Then find the difference (2986 - twice \ the width) and that would be the length. Then multiply length times width \ and get area. Sounds complicated, but really it isn't. Let's look at the \ table.\n"], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[TextData["\nWidth Width Length Area"], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[OutputFormData["\<\ ; ] \:ffff\:ffff\.01\[Florin]roup; endGroup; pictureWidth = 371; pictureHeight = \ 229; preserveAspect; ] \:ffff\:ffff\.01tbold, nohscroll, whiteBox; \tfontset = text, \"C&M\", 12, L\:ffff\.01-$ohscroll; \tfontset = smalltext, \"Chicago\", 12, L2, \ B\:ffff\.01,\[CapitalIDoubleDot]5, nohscroll; \tfontset = input, \"Courier\", 12, L2, bold; \tfontset = output, \"Courier\", 12, L2; \tfontset = message, \"Courier\", 12, L2, R65535, \ nowordwrap\:ffff\.01,Pontset = print, \"Courie\ \>", "\<\ 350 350 2286. 800100. 355 355 2276. 807980. 360 360 2266. 815760. 365 365 2256. 823440. 370 370 2246. 831020. 375 375 2236. 838500. 380 380 2226. 845880. 385 385 2216. 853160. 390 390 2206. 860340. 395 395 2196. 867420. 400 400 2186. 874400.\ \>"], "Output", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "\nNow we can look at the table and see the area closest to 832 000. The \ approximate solution is 370 by 2246."], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False]}, Open]], Cell[CellGroupData[{Cell[TextData[" Solve by graphing two equations\n"], "Section", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "If we used two variables to represent the problem, say x for the width and y \ for the length, the two equations would be: \nxy = 832 000 and \ y + 2x = 2986\nLet's graph those two equations."], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData[ "Clear[x,f,g]\nf[x_]=832000/x\ng[x_]=2986-2x\nPlot[{f[x],g[x]}, \ {x,.0001,2000},PlotStyle->{RGBColor[1,0,0],RGBColor[0,1,0]}]\n\n"], "Input", Editable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart % Scaling calculations 0.02381 0.00048 0.0266 1e-05 [ [(500)] 0.2619 0.0141 0 1 Msboxa [(1000)] 0.5 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The red line shows all the points where the area is 832 000 \ square meters. You write down what the green line represents. Consult with \ your partner.\n\nWe now need to see where both constraints have been met. \ This graph doesn't help us see that very well. Why?"], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "\nDid you decide that the range and domain aren't appropriate for seeing \ where the intersection point(s) are? You're right."], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False]}, Open]], Cell[TextData["\nLet's change them."], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[TextData[{ StyleBox["\n", Editable->False, AspectRatioFixed->False, FontFamily->"Chicago"], StyleBox[ "Plot[{f[x],g[x]}, \ {x,1,1200},PlotRange->{0,3500},PlotStyle->{RGBColor[1,0,0],RGBColor[0,1,0]}]", Editable->False, AspectRatioFixed->False, FontFamily->"Chicago", FontWeight->"Plain"], StyleBox["\n", Editable->False, AspectRatioFixed->False, FontFamily->"Chicago"] }], "Input", Editable->False, AspectRatioFixed->False], Cell[CellGroupData[{Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart % Scaling calculations 0.02381 0.00079 0 0.00018 [ [(200)] 0.18254 -0.0125 0 1 Msboxa [(400)] 0.34127 -0.0125 0 1 Msboxa [(600)] 0.5 -0.0125 0 1 Msboxa [(800)] 0.65873 -0.0125 0 1 Msboxa [(1000)] 0.81746 -0.0125 0 1 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Is our orginal problem represented in \ this picture? Can you give an estimate of the maximum area the farmer could \ have with the fencing he has?"], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False]}, Open]], Cell[CellGroupData[{Cell[TextData[" Further Applications\n"], "Section", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "Did you notice that the previous two functions gave us pictures of \ parabolas? It's probably true that we can vary the other variable (the \ length of the fence) and get other equations which will graph a variable. \ Let's look at some and think about what they mean.\n\nSuppose the farmer has \ half as much fencing. what would the equation be?\n"], "Text", Editable->False, Evaluatable->False, AspectRatioFixed->False], Cell[TextData[ "It would be y = x(1493 - 2x). Let's graph this equation with the \ previous one. 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