(*********************************************************************** 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). 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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[ 464064, 19176]*) (*NotebookOutlinePosition[ 465118, 19212]*) (* CellTagsIndexPosition[ 465074, 19208]*) (*WindowFrame->Normal*) Notebook[{ Cell["", "Input", AspectRatioFixed->True], Cell["", "Input", AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Mini Project -2\nEllipses"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[" by Shirley Treadway"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Introduction"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "One of the more important conic sections is called an ellipse. If it is \ located on the coordinate axes so that the center is at the origin it has a \ very nice equation, (x\[Currency]/a\[Currency]) + (y\[Currency]/b\[Currency]) \ = 1 or (x\[Currency]/b\[Currency]) + (y\[Currency]/a\[Currency])= 1. There is \ a special relationship between a and b and another variable usually referred \ to as c, that is that a\[Currency]= b\[Currency] + c\[Currency]. There are \ two points on the x-axis that have coordinates (c,0) and (-c,0) or in the \ second case on the y-axis with coordinates (0,c) and (0,-c). We would like to \ investigate the relationship between the distances from any point on the \ ellipse, which we will refer to as P, and the two points, (c,0) and (-c,0) or \ (0,c) and (0,-c). 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Now let's look to see if this is just a special case or if this \ observation is true for all ellipses. So that our graphs are manageable let \ me suggest that you go back to the beginning and \nlet a =5 and b = \ 3\[EAcute]", Evaluatable->False, AspectRatioFixed->True], StyleBox["3.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Monaco"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "5. Repeat the activations from above for these values of a and record these \ values on the Investigation Sheet. Be sure to also use the \"Table\" command \ too. \n\n6. What similarity do you see this time? Is it the same as last?\n\n\ 7. If you wish try again with new values of a and b go ahead-- but be sure to \ make a larger than b."], "Text", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Investigate Ellipse (major axis---y-axis)"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "l. 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Activate the Plotall command for 0 and also for `/4. Be sure to take note \ of the differences between this and those in the first set of exercises."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ " \n 4. Please activate the \"Table\" command(below this text) and scroll \ through each graph to investigate the values of the distance from P to \ (0,c1) and the distance from P to (0,c2). This will give the values of \ points on around the circle to 2 `. In each case, please give the value of \ the sum of the two distances on your Investigation Sheet. \n 5. Does this \ number have any similarity to any other given value?\n \n "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["{c1,c2} = c/.Solve[a^2 == b^2 + c^2,c]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {-2*3^(1/2), 2*3^(1/2)}\ \>", "\<\ {-2 Sqrt[3], 2 Sqrt[3]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "Clear[c1plot,c2plot,c1label,c2label,c1val,c2val]\n\nc1plot[t_] := \ Graphics[{RGBColor[1,0,0],Line[{{0,c1},{x[t],y[t]}}]}]\nc2plot[t_] := \ Graphics[{Blue,Line[{{0,c2},{x[t],y[t]}}]}]\nc1label[t_] := \ Graphics[Text[\"c1\",\n {-.2,c1}]]\nc2label[t_] := \ Graphics[Text[\"c2\",\n {0,c2}]]\n\nc1val[t_] := \ Graphics[\n Text[\"= d(f1,p)\" N[dist[{0,c1},{x[t],y[t]}]],\n \ {3a/4,a+3/2}]];\nc2val[t_] := Graphics[\n \ Text[\"= d(f2,p)\"\n N[dist[{0,c2},{x[t],y[t]}]],\n \ {-3a/4,a+3/ 2}]];\n \nctotal[t_] := \ Graphics[\n Text[\"= d(f1,p) + d(f2,p)\" \ N[dist[{0,c2},{x[t],y[t]}] + \n 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Now let's look to see if this is just a special case or if this \ observation is true for all ellipses. So that our graphs are manageable let \ me again suggest that you go back to the beginning \n and let a =5 and b = 3\ \[EAcute]", Evaluatable->False, AspectRatioFixed->True, AnimationDisplayTime->0.5, AnimationCycleOffset->-0.6875, AnimationCycleRepetitions->Infinity], StyleBox["3.\n ", Evaluatable->False, AspectRatioFixed->True, AnimationDisplayTime->0.5, AnimationCycleOffset->-0.6875, AnimationCycleRepetitions->Infinity, FontFamily->"Monaco"] }], "Text", Evaluatable->False, AspectRatioFixed->True, AnimationDisplayTime->0.5, AnimationCycleOffset->-0.6875, AnimationCycleRepetitions->Infinity], Cell[TextData[ "7. Repeat the activations from above for these values of a and b and record \ these values on the Investigation Sheet. Be sure to also use the \"Table\" \ command too. \n\n8. What similarity do you see this time? Is it the same as \ last?\n\n9. 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