(*********************************************************************** 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[ 8580, 316]*) (*NotebookOutlinePosition[ 9216, 339]*) (* CellTagsIndexPosition[ 9172, 335]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Examples from Section 6.5 Exponential Growth and Decay\ \>", "Section"], Cell["p. 488", "Subsection"], Cell[CellGroupData[{ Cell["Growth of Bacteria", "Subsection"], Cell[CellGroupData[{ Cell["8.", "Subsubsection"], Cell[TextData[{ "A colony of bacteria is grwon under ideal conditions in a laboratory so \ that the population increases exponentially with time. At the end of 3 h \ therre are 10,000 bacteria. At the end of 5 h there are 40,000. How many \ bacteria were present initially.\n\n", StyleBox["Solution.", FontSlant->"Italic"], " The number of bacteria is given by ", Cell[BoxData[ \(y\ = \ \(y\_0\) e\^\(k\ t\)\)]], "where ", Cell[BoxData[ \(y\_0\)]], "is the initial amount and ", Cell[BoxData[ \(k\)]], " is some constant and ", Cell[BoxData[ \(t\)]], " is in hours. To determine ", Cell[BoxData[ \(k\)]], " we use the two given points of data:\n", Cell[BoxData[ \(10000\ = \ \(y\_0\) e\^\(k\ 3\)\)]], "\n", Cell[BoxData[ \(40000\ = \ \(y\_0\) e\^\(k\ 5\)\)]], "\nDivide the second equation by the first to get\n", Cell[BoxData[ \(4\ = \ \(e\^\(k \((5\ - \ 3)\)\) = \ e\^\(2\ k\)\)\)]], ".\nTaking logarithms gives\n", Cell[BoxData[ \(ln \((4)\)\ = \ 2\ k\)]], " or ", Cell[BoxData[ \(k\ = \ \(\(1\/2\) ln \((4)\)\ = \ ln \((2)\)\)\)]], "\nTherefore,\n", Cell[BoxData[ \(y\ = \ \(y\_0\) e\^\(ln \((2)\) t\)\)]], " = ", Cell[BoxData[ \(\(y\_0\) 2\^t\)]], ".\nWe can now determine ", Cell[BoxData[ \(y\_0\)]], " by plugging in one of the original data points:\n", Cell[BoxData[ \(10000\ = \ y\_0\ 2\^3\)]], "\nwhich gives ", Cell[BoxData[ \(y\_0 = \ \(10000/8\ = \ 12500\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[{10000\ == \ y\_0\ E\^\(k\ 3\), 40000\ == \ y\_0\ E\^\(k\ 5\)}, \ {k, \ y\_0}]\)], "Input"], Cell[BoxData[ \(Solve::"ifun" \( : \ \) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message"], Cell[BoxData[ \(Nonreal::"warning" \( : \ \) "Nonreal number encountered."\)], "Message"], Cell[BoxData[ \({{y\_0 \[Rule] \(-1250\), k \[Rule] Nonreal}, {y\_0 \[Rule] 1250, k \[Rule] Log[2]}}\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Radioactive Decay", "Subsection"], Cell[CellGroupData[{ Cell["17.", "Subsubsection"], Cell[TextData[{ "The decay equation for radon-222 gas is known to be ", Cell[BoxData[ \(y \((t)\)\ = \ \(y\_0\) e\^\(\(-0.18\) t\)\)]], " with ", Cell[BoxData[ \(t\)]], " in days. About how long will it take the radon in a sealed sample of air \ to fall to 90% of its original value?\n\n", StyleBox["Solution.", FontSlant->"Italic"], " We must solve for ", Cell[BoxData[ \(t\_1\)]], " in the equation:\n", Cell[BoxData[ \(0.90\ y\_0\ = \ \ y \((t\_1)\)\)]], " = ", Cell[BoxData[ \(\(y\_0\) e\^\(\(-0.18\) t\_1\)\)]], ".\nThe ", Cell[BoxData[ \(y\_0\)]], "'s cancel and we take logarithms to get\n", Cell[BoxData[ \(ln \((0.90)\)\ = \ \(-0.18\)\ t\)]], " or ", Cell[BoxData[ \(t\ = \ \(\(-ln\) \((0.90)\)/0.18\ = \ 0.585336198099034898`\)\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[0.90\ == E\^\(\(-0.18\) t\)]\)], "Input"], Cell[BoxData[ \(Solve::"ifun" \( : \ \) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message"], Cell[BoxData[ \({{t \[Rule] 0.585336198099034898`}}\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Newton's Law of Cooling", "Subsection"], Cell[CellGroupData[{ Cell["23.", "Subsubsection"], Cell[TextData[{ "A pan of warm water (46\[Degree]C) was put in a refrigerator. Ten minutes \ later the water's temperature was 39\[Degree]C; 10 minutes after that, it was \ 33\[Degree]C. Use Newton's Law of Cooling to estimate how cold the \ refrigerator was.\n\n", StyleBox["Solution.", FontSlant->"Italic"], " The equation for the temperature ", Cell[BoxData[ \(T \((t)\)\)]], " at time ", Cell[BoxData[ \(t\)]], " (in minutes) is given on p.487:\n", Cell[BoxData[ \(T \((t)\)\ - \ T\_s\ = \ \((T\_0\)\)]], " - ", Cell[BoxData[ \(\(T\_s)\) e\^\(\(-k\)\ t\)\)]], ",\nwhere ", Cell[BoxData[ \(T\_s\)]], " is the temperature of the refrigerator, ", Cell[BoxData[ \(T\_0\)]], " is the initial temperature of the water (46\[Degree]C), and ", Cell[BoxData[ \(k\)]], " is some constant. Plugging in the given data ", Cell[BoxData[ \(T \((10)\)\ = \ 39\)]], " and ", Cell[BoxData[ \(T \((20)\)\ = \ 33\)]], " gives two equations:\n", Cell[BoxData[ \(39\ - \ T\_s = \ \((46\ - \ T\_s)\) e\^\(\(-k\)\ 10\)\)]], ",\n", Cell[BoxData[ \(33\ - \ T\_s = \ \((46\ - \ T\_s)\) e\^\(\(-k\)\ 20\)\)]], ".\nSolving for ", Cell[BoxData[ \(k\)]], " in both equations gives:\n", Cell[BoxData[ \(k\ = \ \(\(-\(1\/10\)\) ln \((\(39 - T\_s\)\/\(46 - T\_s\))\)\ = \ \(-\(1\/20\)\) ln \((\(33 - T\_s\)\/\(46 - T\_s\))\)\)\)]], "\nThe last equation can be exponentiated to yield\n", Cell[BoxData[ \(\(\(\((\(39 - T\_s\)\/\(46 - T\_s\))\)\^\(1/10\) = \)\ \)\)]], Cell[BoxData[ \(\((\(33 - T\_s\)\/\(46 - T\_s\))\)\^\(1/20\)\)]], "\n(note that we could have arrived at this point by isolating ", Cell[BoxData[ \(e\^\(-k\)\)]], "in the original equations) or\n", Cell[BoxData[ \(\(\(\((\(39 - T\_s\)\/\(46 - T\_s\))\)\^2 = \)\ \)\)]], Cell[BoxData[ \(\((\(33 - T\_s\)\/\(46 - T\_s\))\)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[ \((\(39 - T\_s\)\/\(46 - T\_s\))\)\^2 == \ \((\(33 - T\_s\)\/\(46 - T\_s\))\)]\)], "Input"], Cell[BoxData[ \({{T\_s \[Rule] \(-3\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "So the temperature of the refrigerator was ", Cell[BoxData[ \(\(-3\)\)]], ". The constant ", Cell[BoxData[ \(k\)]], " is\n", Cell[BoxData[ \(k\ = \ \(\(-\(1\/10\)\) ln \((\(39 - \((\(-3\))\)\)\/\(46 - \((\(-3\))\)\))\)\ = \ \(-ln\) \((6\/7)\)\)\)]], " = ", "0.0154151" }], "Text"], Cell[TextData[{ "The tempature equation now reads\n", Cell[BoxData[ \(T \((t)\)\ - \ \((\(-3\))\)\ = \ \((46\)\)]], " - ", Cell[BoxData[ \(\(\((\(-3\))\))\) e\^\(t 1\/10\ ln \((6\/7)\)\)\ = \ 49\ \((6\/7)\)\^\(t/10\)\)]], "\n", Cell[BoxData[ \(\(T \((t)\)\ = \ \(\(-3\)\ + \)\ \)\)]], Cell[BoxData[ \(49\ \((6\/7)\)\^\(t/10\)\)]], "." }], "Text"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, WindowSize->{520, 600}, WindowMargins->{{Automatic, 133}, {37, Automatic}} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 81, 3, 72, "Section"], Cell[1815, 56, 28, 0, 45, "Subsection"], Cell[CellGroupData[{ Cell[1868, 60, 40, 0, 45, "Subsection"], Cell[CellGroupData[{ Cell[1933, 64, 27, 0, 42, "Subsubsection"], Cell[1963, 66, 1562, 53, 311, "Text"], Cell[CellGroupData[{ Cell[3550, 123, 126, 2, 31, "Input"], Cell[3679, 127, 154, 3, 39, "Message"], Cell[3836, 132, 93, 1, 23, "Message"], Cell[3932, 135, 127, 2, 27, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[4120, 144, 39, 0, 45, "Subsection"], Cell[CellGroupData[{ Cell[4184, 148, 28, 0, 42, "Subsubsection"], Cell[4215, 150, 853, 30, 158, "Text"], Cell[CellGroupData[{ Cell[5093, 184, 68, 1, 31, "Input"], Cell[5164, 187, 154, 3, 39, "Message"], Cell[5321, 192, 69, 1, 27, "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[5451, 200, 45, 0, 45, "Subsection"], Cell[CellGroupData[{ Cell[5521, 204, 28, 0, 42, "Subsubsection"], Cell[5552, 206, 1985, 61, 338, "Text"], Cell[CellGroupData[{ Cell[7562, 271, 133, 3, 47, "Input"], Cell[7698, 276, 57, 1, 27, "Output"] }, Open ]], Cell[7770, 280, 366, 15, 73, "Text"], Cell[8139, 297, 401, 14, 78, "Text"] }, Open ]] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)