.MCAD 308000000 \  docDocumentMmcObject[    кк  dim_formatTmasslengthtimecharge temperature luminosity substancegNumericalFormatQdii shpRectVmcDocumentObjectState\Ё┤ mcPageModelKа?а?А?А?mcHeaderFooterI@I(Created: Sept. 1997|NUpdated: Sept. 1997|FAuthor:Peter J. Hansen|Npage |P CHeaderFooterJ@ъ{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq15 Arial;}{\f3\fswiss\fprq12 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\plain\f2\fs18 \par } @э{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq15 Arial;}{\f3\fswiss\fprq12 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qc\plain\f2\fs18 \par } @э{\rtf1\ansi\ansicpg1252\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq15 Arial;}{\f3\fswiss\fprq12 Arial;}} {\colortbl\red0\green0\blue0;} \deflang1033\pard\qr\plain\f2\fs18 \par } @J'Created: Sept. 1997 Updated: Sept. 1997{f}Author:Peter J. Hansen page {n}@J@J№йё╥MbP?№йё╥MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultW?Normal        Аfont_style_listO font_stylePє   Ў         VariablesArial@Pє   Ў         ConstantsArial@Pє   Ў        TextArial@Pє   Ў        Greek VariablesSymbol@Pє   Ў        User^1Arial@Pє   Ў        User^2 Courier New@Pє   є        User^3System@Pє   Ў         User^4Script@Pє   Ў         User^5Roman@Pє   Ў         User^6Modern@Pє   Ў        User^7Times New Roman@Pє   Ў        SymbolsSymbol@Pє   є        Current Selection FontArial@Pє   є        Undefined Font@Pє   є        HeaderArial@PЎ   Ў       FooterArial@Pє   Ў        Rotated Math FontArialX TextRegion* docRegionGshpBoxU`┴,`(l@aa CharacterMap-RangeMap;'Propagation of Error: Mathcad Template ChrPropMap7' RangeElem< ChrPropData8 RangeData=|0,0,128 < 8|0,0,128  ParPropMap9' <'  ParPropData:@W,            EmbedMap1 <LinkMap/'<'LinkData0    @NormalArial    *@U└CPг└Pэ@Р`Р`-@zPeter J. Hansen Department of Chemistry Northwestern College 101 7th Street SW Orange City, IA 51041 pjhansen@nwciowa.edu7z8,0,0,128;?<@@8,_0,0,128=?-9╫@A<╫@B:@W,            1@C</╫@D<╫@E0    @NormalArial    @F*@U╖(─└Йь  -0How do we calculate the uncertainty in the area?7090@G<0@H:@W,            1@I</0@J<0@K0    @NormalArial    @L*@U╫ёрЛ-@зQuestion: The sum of the uncertainties in the height and width is clearly not equal to the uncertainty in the area. Why is this obvious? Hint: consider their units.7N з@M< @N8@L}255,0,0@O@@dA=\dAA?@@дA=\dwA@@@дA<hAA*@Uh█ ыhшйве -(calculate partial derivatives)79AB<AC:@W,            1AD</AE<AF0    @NormalArial    AG@B@Up¤Ч!AH@@ pAI@@,┼АAHAJ@@№@AIAK@@Оp@AJAL@@¤АAKAM@@dAL\eAN@@дALAAO@@┤AJ2AP@@Й╟АAIAQ@@╩@APAR@@№@AQAS@@dARwAT@@┤AR2AU@@№АAQAV@@Оp@AUAW@@¤АAVAX@@dAW\eAY@@дAWhAZ@@┤AU2A[@@╩АAPA\@@№@A[A]@@dA\hA^@@┤A\2A_@@№АA[A`@@Оp@A_Aa@@¤АA`Ab@@dAa\eAc@@дAawAd@@┤A_2Ae*@U``1╣╣╣-$(substitute for partial derivatives)7$9$Af<$Ag:@W,            1Ah</$Ai<$Aj0    @NormalArial    Ak@B@UH%MAo8%Al@@ pAm@@,┼АAlAn@@№@AmAo@@Оp@AnAp@@¤АAoAq@@dAp\eAr@@дApAAs@@┤An2At@@Й╟АAmAu@@╩@AtAv@@№@AuAw@@Оp@AvAx@@┤Aw15.73Ay@@┤Av2Az@@№АAuA{@@Оp@AzA|@@┤A{0.10A}@@┤Az2A~@@╩АAtA@@№@A~AА@@Оp@AAБ@@┤AА3.47AВ@@┤A2AГ@@№АA~AД@@Оp@AГAЕ@@┤AД0.15AЖ@@┤AГ2AЗ*@Up+ ;p8/ЯЫЬ -(substitute numeric values)79AИ<AЙ:@W,            1AК</AЛ<AМ0    @NormalArial    AН@B@UИMiп`'AО@@ pAП@@,┼АAОAР@@№@AПAС@@Оp@AРAТ@@¤АAСAУ@@dAТ\eAФ@@дAТAAХ@@┤AР2AЦ@@Й╟АAПAЧ@@tAЦ2.474AШ@@┤AЦ0.2709AЩ*@UаS╘cа`_╕44 - (simplify)7 9 AЪ< AЫ:@W,            1AЬ</ AЭ< AЮ0    @NormalArial    AЯ*@Us│Ае @4-A'Note, although the uncertainty in the width (ew = 0.15) is 50% greater than the uncertainty in the height (eh = 0.10), the latter makes a much greater contribution to the uncertainty in the area (eA). Error propagation calculations help one distinguish between major and minor sources of error.7╞.'Aа<-Aб8AЯ0,0,128Aв<Aг8AЯSymbol0,0,128AаAд<Aе8AЯ_0,0,128AвAж<<Aз8AЯ0,0,128AдAи<Aй8AЯSymbol0,0,128AжAк<Aл8AЯ_0,0,128AиAм4└.(-&Final result: Area = 54.6 ▒ 1.6 cm27%%&A╪<%A┘8A╫} Courier New0,0,0A┌<A█8A╫] Courier New0,0,0A╪A┌A╪9&A▄<&A▌:@W,            1A▐</&A▀<&Aр0    @NormalArial    Aс*@U#C0F   -@ЮBy convention, the uncertainty in the final result is rounded to two significant digits and the result itself is rounded to the same number of decimal places.7Ю9ЮAт<ЮAу:@W,            1Aф</ЮAх<ЮAц0    @NormalArial    Aч PageBreakE@UX@`XйAш*@UБ,ФР¤-@G ====================== Calculation Template ======================7.GAщ<Aъ8Aш0,0,128Aы<Aь8Aш| 0,0,128AщAэ<Aю8Aш0,0,128AыAэAы9GAя@@Оp@B=B?@@√АB>B@@@dB?\dzBA@@дB?\dxBB@@┤B=2BC@@№АB@@№АC7C?@@dC>\emC@@@┤C>2CA@B@U(r kД QА °CB@@ pCC@@М┴АCBCD@@dCCterm1CE@@Ц─АCCCF@@+@CESerial_DisplayNodeXCG@@дCE _n_u_l_l_CH@B@U(Ф б┼ R░ ∙CI@@ pCJ@@ ┴АCICK@@dCJterm2CL@@╩АCJCM@@№@CLCN@@Оp@CMCO@@√АCNCP@@ХK@COCQ@@дCPmCR@@№АCOCS@@dCRVCT@@┤CR2CU@@┤CM2CV@@№АCLCW@@dCV\eVCX@@┤CV2CY@B@U(╥ kф Qр №CZ@@ pC[@@М┴АCZC\@@dC[term2C]@@Ц─АC[C^@@+@C]@XC_@@дC] _n_u_l_l_C`*@Uє   c  Ї-@TNotice that the volume error term is over 400 times as large as the mass error term.7T9TCah Cu@@ pCv@@ ┴АCuCw@@dCv\edCx@@Ъ{АCvCy@@Й╟АCxCz@@dCyterm1C{@@дCyterm2C|@B@U(z WМ =И ZC}@@ pC~@@М┴АC}C@@dC~\edCА@@Ц─АC~CБ@@+@CА@XCВ@@дCА _n_u_l_l_CГ@B@U(Ъ |м bи XCД@@ pCЕ@@М┴АCДCЖ@@╬@CЕCЗ@@dCЖdCИ@@ОpАCЖCЙ@@ ├АCИCК@@dCЙmCЛ@@дCЙVCМ@@Ц─БCЕCН@@+@CМ@XCО@@дCМ _n_u_l_l_6JCП*@U╗ ы ╚ ш  0 0-@тStep #8. Round the uncertainty to two significant digits, and then round the result to as many decimal places as the uncertainty. Use [Alt] [0177] (with Num Lock ON) from the number keypad to produce the plus/minus sign (▒).7ФтCР<CС8CПu0,0,128CТ@@dD=VD?@@┤D=2D@@@┤D82DA@@√АD7DB@@№@DADC@@Оp@DBDD@@дDC\eVDE@@┤DB2DF@@№АDADG@@Оp@DFDH@@√АDGDI@@dDHmDJ@@дDHVDK@@┤DF2DL@B@U(╝┴эB╪ЕDM@@ pDN@@,┼АDMDO@@√@DNDP@@№@DODQ@@dDP\edDR@@┤DP2DS@@№АDODT@@dDSdDU@@┤DS2DV@@√АDNDW@@Оp@DVDX@@Й╟АDWDY@@╩@DXDZ@@№@DYD[@@dDZ\emD\@@┤DZ2D]@@№АDYD^@@dD]VD_@@┤D]2D`@@╩АDXDa@@№@D`Db@@dDamDc@@┤Da2Dd@@№АD`De@@dDd\eVDf@@┤Dd2Dg@@ОpАDVDh@@╩АDgDi@@№@DhDj@@dDimDk@@┤Di2Dl@@№АDhDm@@dDlVDn@@┤Dl2Do*@U(√8 (З-2The latter can be further simplified (by hand) to:7292Dp<2Dq:@W,            1Dr</2Ds<2Dt0    @NormalArial    Du@B@U(╡HP8ЙDv@@ pDw@@,┼АDvDx@@№@DwDy@@Оp@DxDz@@√АDyD{@@dDz\edD|@@дDzdD}@@┤Dx2D~@@Й╟АDwD@@№@D~DА@@Оp@DDБ@@√АDАDВ@@dDБ\emDГ@@дDБmDД@@┤D2DЕ@@№АD~DЖ@@Оp@DЕDЗ@@√АDЖDИ@@dDЗ\eVDЙ@@дDЗVDК@@┤DЕ2DЛ*@U([{(hМЁЁ Ё -@ФThis result indicates that the square of the relative error in density is equal to the sum of the squares of the relative errors in mass and volume.7А-ФDМ<-DН8DЛ0,0,0DО<DП8DЛw0,0,0DМDР<6DС8DЛ0,0,0DОDТ<DУ8DЛw0,0,0DРDФ<DХ8DЛ0,0,0DТDФDО9ФDЦ<ФDЧ:@W,            1DШ</ФDЩ<ФDЪ0    @NormalArial    DЫ*@U(г (░Сц\ц\-A╜ ЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧ Acknowledgment: PJH acknowledges the National Science Foundation for support of the 1997 NSF-UFE Workshop on "Numerical Methods in the Undergraduate Chemistry Curriculum Using the Mathcad Software" and its organizers Jeff Madura, Andrzej Wierzbicki and Sidney Young, Department of Chemistry, University of South Alabama, Mobile, AL 36688-0002. ЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧЧ7╢╜DЬ<DЭ8DЫ~ 0,0,0DЮ<DЯ8DЫ~ 0,0,0DЬDа<Dб8DЫ~ 0,0,0DЮDв<Dг8DЫ~ 0,0,0DаDд<Dе8DЫ~ 0,0,0DвDж<Dз8DЫ0,0,255DдDи< Dй8DЫ~ 0,0,0DжDк<Dл8DЫt Times New Roman0,0,0DиDм<Dн8DЫ~ 0,0,0DкDо