Table11.6:BasicSketchCriteria
sketchedfunctionmustpassthroughthepoint(xcoord,
ycoord)
goes_through(xcoord, ycoord)
sketchedfunctionmustbeincreasing
increasing
sketchedfunctionmustbedecreasing
decreasing
sketchedfunctionmustbeconcaveup
concave_up
sketchedfunctionmustbeconcavedown
concave_down
sketchedfunctionmustbelinear
linear
Torestrictacriteriontoaninterval[a,b],use\check[a,b]{criterion}.Thisoptionisignoredfor
goes_through.
Thecriterioncanalsobeabooleanexpression;usefulstatementsinthiscaseare:
Table11.7:StatementsforAdditionalSketchCriteria
returntheslopeofthecurveatthex-valuexval.
slope_at(xval)
returnthey-valueforthex-valuexval.
value(xval)
returntrueifexpr1isequaltoexpr2.Otherwise,it
returnsfalse.
expr1 == = expr2
returntrueifexpr1islessthanexpr2.Otherwise,it
returnsfalse.
expr1 < < expr2
returntrueifexpr1isgreaterthanexpr2.Otherwise,
itreturnsfalse.
expr1 > > expr2
booleanconstant
true
booleanconstant
false
Forexample,value(xcoord) == ycoordisequivalenttogoes_through(xcoord, ycoord).
Morethanone\checkstatementcanappear,inwhichcaseeverycriterionmustbesatisfied.
Syntax
\begin{sketch}[gridlines option]{xMin}{xMax}{yMin}{yMax}
\example{<point-list>}
\check{criterion1}
\check{criterion2}
\check{criterion3}
\end{sketch}
Example1
\begin{question}{sketch}
\qutext{Sketch the graph of the function n $y=x+1$.}
\begin{sketch}[4]{-2}{2}{-2}{2}
\example{-2,-1
1,2}
\check{linear}
\check{goes_through(0,1)}
\end{sketch}
\end{question}
Example2
Thisexampleusesrandomizedvariables.
\begin{question}{sketch}
\code{$a = = int(rand(2,6));
$a2
= int($a*$a);
$xmax
= int(2*$a);
11.6AdvancedQuestionTypesinLaTeX • • 281
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$x
= sqrt(2)*$a;}
\qutext{Sketch the graph of the function n $y=x^2/\var{a2}$.}
\begin{sketch}[4]{-\var{xmax}}{\var{xmax}}{-4}{4}
\example{-\var{x},2
-\var{a},1
0,0
\var{a},1
\var{x},2}
\check{goes_through(0,0)}
\check{goes_through(-\var{a},1)}
\check{goes_through(\var{a},1)}
\check{slope_at(0)
== 0}
\check[-\var{x},0]{decreasing}
\check[0,\var{x}]{increasing}
\check{concave_up}
\end{sketch}
\end{question}
IncludingGraphicReferencesinLaTeXQuestions
IncludingGraphicsUsing\image
YoucaninsertanimageinaLaTeXquestion.The\imagemacroistherecommendedmethodforincludinggraphics
inaquestion.IfthefileisavailablelocallyasanEncapsulatedPostscript(epsorepsf)file,itisincluded(\image
calls\epsfigtoincludethegraphic).
Important:Donotexplicitlycall\epsfig.Using\epsfigdirectlydoesnotensurethatproperMapleT.A.code
results.
Toincludegraphics:
1. SelectalocationinyourMapleT.A.classwebsiteforgraphicsfiles.Itisrecommendedthatyouuseadistinct
folderfortheimagesineachquestionbank.UploadthegraphictoafolderonyourMapleT.A.classwebsite.For
informationonuploading,seeUploadingImagesandReferenceFilestotheClassWebSite(page88).Afterup-
loadinggraphicsfiles,youcanclicktheiconbesideafiletoseeitsfullpath,whichwillhavetheform
http://.../web/<class>/Public_Html/<folder>/...
where<class>isMapleT.A.'sinternalidentifierforyourclass,and<folder>isthefolderthatcontainstheimages.
2. Insertthe\setImageBasedirectivenearthetopoftheLaTeXfiletospecifythepathyoudeterminedinstep1.
Forexample,ifyourMapleT.A.classidentifierismyclass,andyourgraphicsfolderisImages,usethedirective:
\setImageBase{web/myclass/Public_Html/Images}
3. Includegraphicsinaquestionusingthe \imagemacro.Theformatforthe\imagemacrois:
\image[extension]{filename}
Theextensionargumentisoptional.Itspecifiestheextensiontobeappendedtothefilename(thesecondargument)
whenthequestionbankisconvertedtoMapleT.A.format.Thedefaultvalueisjpg.
Ifthefilenameargumentisafilenamewithanextension,thefirstargumentisignored.Ifthefilenameargument
isafilenamewithoutanextension,the.epsextensionisusedinpreviewmode,andextension(orthedefault
.jpg)isusedinproductionprocessing.
Sample1
\image{graphic}
Placesthefilegraphic.epsfromthecurrentdirectoryinthedvioutputinpreviewmode.Inproduction
processing,itproducesanimagetagpointingtographic.jpginthedirectoryspecifiedby\setImage-
Base.
282 • 11AuthoringQuestionsinLaTeX
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Sample2
\image[gif]{graphic}
Placesthefilegraphic.epsfromthecurrentdirectoryinthedvioutputinpreviewmode.Inproduction
processing,itproducesanimagetagpointingtographic.gifinthe \setImageBasedirectory.
Sample3
\image{graphic.pct}
Doesnotplacethefilegraphic.pctinthedvioutput.Inproductionprocessing,itproducesanimage
tagpointingtographic.pctinthe\setImageBasedirectory.
Notes
• Whilethe\epsfigmacroallowsvariousoptions,forexample,width,\imagedoesnotsupportanyoptions.
EncapsulatedPostscriptfilesappearatfullsizeinthedvifile.
• Theconversiontoolsdonotprovidegraphicconversionservices.Youmustusethird-partytoolstoproduce
webimagefiles.
Example
Note:forthisexampletoworkasis,youmustfirstuploadaGIFimagequad.giftoyourMapleT.A.classwebsite.
YouwillalsoneedtosettheimagebaseforyourLaTeXfileasdescribedabove.
\begin{question}{Numeric}
\qutext{Quadrilateral $ABCD$ $ is circumscribed d about a circle, as shown, with h $AB$=\var{a} } cm,
and $CD$=\var{b} cm. . Find the e perimeter of the quadrilateral.
\image{quad.gif}}
\code{$a=range(5,25);
$b=range($a+1,$a+5);
$ans=2*(($a)+($b));}
\answer{$ans}[cm]
\end{question}
LabelingGraphicsinLaTeXQuestions
LabeledGraphics-labelgraphicEnvironment
Thelabelgraphicenvironmentallowsyoutoaddtextlabelstographics(gifandjpgfiles).
ToaddtextlabelstographicsinyourLaTeXquestion:
1. SelectalocationinyourMapleT.A.classwebsiteforgraphicsfiles.Itisrecommendedthatyouuseadistinct
folderfortheimagesineachquestionbank.UploadthegraphictoafolderonyourMapleT.A.classwebsite.For
informationonuploading,seeUploadingImagesandReferenceFilestotheClassWebSite(page88).Afterup-
loadinggraphicsfiles,youcanclicktheiconbesideafiletoseeitsfullpath,whichwillhavetheform
http://.../web/<class>/Public_Html/<folder>/...
where<class>isMapleT.A.'sinternalidentifierforyourclass,and<folder>isthefolderthatcontainstheimages.
2. Insertthe\setImageBasedirectivenearthetopoftheLaTeXfiletospecifythepathyoudeterminedinstep1.
Forexample,ifyourMapleT.A.classidentifierismyclass,andyourgraphicsfolderisImages,usethedirective:
11.6AdvancedQuestionTypesinLaTeX • • 283
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\setImageBase{web/myclass/Public_Html/Images}
3. Toincorporatetheimagefileintoaquestion,usethelabelgraphicenvironment.The\begin{labelgraph-
ic}commandtakesthreearguments:
i. Thenameoftheimagefile.
ii. Thewidthoftheimagefileinpixels.
iii.Theheightoftheimagefileinpixels.
Todeterminethewidthandheightofthegraphic,useathird-partygraphicsapplication(forexample,Microsoft
Paint).
• Tolabelthegraphic,usethe\grlabelmacro.The\grlabelmacrotakesthreearguments:
i. Thetextofthelabel.
ii. Thex-coordinateofthelabel,inpixels.
iii.They-coordinateofthelabel,inpixels.
• Thex-andy-coordinatesaremeasuredfromtheupperleftcornerofthegraphic.
• Morethanone\grlabelcommandcanbeused.
Note:Withoutany\grlabelcommands,thelabelgraphicenvironmentonlyincorporatestheimagefileintoaquestion.
Toincludeagraphic,itisrecommendedthatyouusethe\imagemacro.Formoreinformationontheimagemacro,seeIncluding
GraphicReferencesinLaTeXQuestions(page282).
Example
Note:Forthisexampletoworkasis,youmustfirstuploada309x256pixelGIFimageLabeledGraphic.gif
toyourMapleT.A.classwebsite.YouwillalsoneedtosettheimagebaseforyourLaTeXfileasdescribedabove.
\begin{question}{Multi Formula}
\qutext{Suppose that you u throw w a a rock from the top p of f a a \var{h} } m m cliff
with a velocity y of f \var{v} m/s s in n the three directions shown. Neglecting
aerodynamic drag, use the e principle of f work and energy to determine the
velocity of the e rock as s it hits the e ground d in n the e upward, horizontal,
and downward angles, , respectively.
\begin{labelgraphic}{LabeledGraphic.gif}{309}{256}
\grlabel{$h m}{80}{174}
\grlabel{$angle}{200}{134}
\grlabel{$angle}{200}{97}
\end{labelgraphic}}
\code{$angle=rand(10,50,2);
$h=rand(7,14,2);
$v=rand(8,17,2);
$ansa=sig(3, sqrt(2*9.8*$h + $v^2));
$ansb=$ansa;
$ansc=$ansa;
$tol=lsu(2, $ansa);}
\answer{($ansa ? $tol)*m/s;($ansb ? ? $tol)*m/s;($ansc c ? $tol)*m/s}
\end{question}
11.7Tutorial:UsingAlgorithmsinLaTeXQuestions
AlgorithmicQuestions
Randomizedvariablescanbeincorporatedintoanyquestion.Rulesandrecommendationsforeffectiveuseofrandomized
variablesaregivenbelow,illustratedbyrelatedexamples.
284 • 11AuthoringQuestionsinLaTeX
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• Thecodethatcreatesandmanipulatestherandomizedvariablesiscontainedinthe\code{}sectionofthe
question.Eachvariableinthecodesectionisdesignatedwitha$symbol,suchas$m,$ans,and$b2.The
firstcharacterafterthe$symbolmustbealetter.Eachcodestatementisoftheform
$variable=expression;
Separatemultiplevariabledefinitionswithasemicolon.
Note:TeXcomments(%)cannotappearintheargumentto\code.Toincludecomments,usethecommentcontrolse-
quence.
Theusualarithmeticoperationsandfunctionscanbeusedintheexpressionportionofthecodestatement.
However,unlikemostotherprogramminglanguages,variablescannotberedefinedintermsofthemselves(for
example,thestatement$a=$a+1;isnotallowed).Foradescriptionoftheadditionalfunctionsthatcanbe
usedinthecodesections,seeFunctionswithinAlgorithms(page239).
• Tousethevariablesinothersectionsofthequestion,enclosetheminthe\varmacro.Forexample,torefer
tothevariable$a,use:
\var{a}
Inthefollowingexamples,aFormulaquestionisdesignedthroughaseriesofrefinements.
Example1
\begin{question}{Formula}
\qutext{Find the solution n of f the linear equation
$\var{a}x+\var{b}=\var{c}$.}
\answer{(\var{c}-\var{b})/\var{a}}
\code{
$a=rint(12)+1;
$b=rint(12)+1;
$c=rint(12)+1;
}
\end{question}
Theabovecoderandomlydeterminesintegercoefficients$a,$b,and$cbetween1and12.Theansweris
calculatedinthe\answersection.
Althoughtheaboveexampledoesnotcauseerrors,thereareseveralsideeffects:
• Theanswerisdisplayedintheform(7-3)/5(forexample).Thisprovidesanindicationofthemethodfor
obtainingtheanswer,whichmaybeinappropriate.
• Theaboveformatcanreturnanswersoftheform(7-7)/5,whichismorecomplicatedthanthesimplified
form,0.
Toavoidtheseissues,rewritethisexampleasfollows,usinganewvariable$ans(sothatyoureturnonly
thefinalanswer).
Example2
RefinementofExample1.
\begin{question}{Formula}
\qutext{Find the solution n of f the linear equation
$\var{a}x+\var{b}=\var{c}$.}
\answer{\var{ans}}
\code{
$a=rint(12)+1;
$b=rint(12)+1;
11.7Tutorial:UsingAlgorithmsinLaTeXQuestions • • 285
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$c=rint(12)+1;
$ans=($c-$b)/$a;
}
\end{question}
Thereisanotherproblem.Thevariable$acanequal1,inwhichcasetheequationhastheform1x+5=8.To
avoidthissituation,imposetheconditionthat$abeanintegerbetween2and12.
Although$ccanbe0ornegative,if$bis0ornegativetheequationisagainunsimplified.Generalizethe
codetoallow$ctobeanyintegerbetween-12and12.
Example3
\begin{question}{Formula}
\qutext{Find the solution n of f the linear equation
$\var{a}x+\var{b}=\var{c}$.}
\answer{\var{ans}}
\code{
$a=rint(11)+2;
$b=rint(12)+1;
$c=rint(25)-12;
$ans=($c-$b)/$a;
}
\end{question}
Thereisanotherproblem.Theanswer$ansisa(potentiallyinexact)floating-pointnumber,forexample,
.666667,insteadof2/3.Tosolvetheproblem,useastringvariabletopreventMapleT.A.fromperforming
afloating-pointevaluation.
Example4
\begin{question}{Formula}
\qutext{Find the solution n of f the linear equation
$\var{a}x+\var{b}=\var{c}$.}
\answer{\var{ans}}
\code{
$a=rint(11)+2;
$b=rint(12)+1;
$c=rint(25)-12;
$num=$c-$b;
$ans="$num/$a";
}
\end{question}
Note:Writing$ans="($c-$b)/$a";displaysanswersoftheform(11-6)/3.First,calculatethenumeratorand
storeitasanintegerin$num,andthencreatethefractionwiththestring"$num/$a".
Thislastchangeintroducesanewproblem.Thesystemdisplayscorrectanswersoftheform10/5or2/4.To
makeanswersdisplayassimplifiedfractions,implementonemorechange.
Example5
Finalrefinementofthequestion.
\begin{question}{Formula}
\qutext{Find the solution n of f the linear equation
$\var{a}x+\var{b}=\var{c}$.}
\answer{\var{ans}}
\code{
$a=rint(11)+2;
$b=rint(12)+1;
$c=rint(25)-12;
$num=$c-$b;
286 • 11AuthoringQuestionsinLaTeX
$ans=frac($num,$a);
}
\end{question}
Thestringreturnedbythefraccommandisareducedfraction.
Example6-QuestionwithaPlot
\begin{question}{Formula}
\qutext{Enter the expression plotted below. Give only the expression itself, omitting
any "y=" or "f(x)=" from your answer. \newline
\var{plot}.}
\answer{x^2+\var{a}x+\var{b}}
\code{
$a=rint(2, 9);
$b=rint(1, 9);
$plot=plotmaple("plot(x^2+$a*x+$b, x=-10..10)");
}
\end{question}
Toaddaplottoaquestioninthequestiontext,feedback,orhints,addanalgorithmicvariableusingtheplot-
maplecommand.Formoreinformation,seeplotmaple(text)(page247).
Usethefollowingguidelinesforalgorithms:
• Ensurethatallpossiblevaluesofrandomizedvariablesarevalidinthequestiontextandtheanswer,andthat
theresponsewillgradecorrectly.
• Definerandomizedvariablessothatyouavoidexpressionslike1x,0x,x+-3,or-3/-4.Ifthesevaluesare
required,writethequestionsothattheydisplayproperly.
• Writethecodesothatifthestudentanswersincorrectly,thecorrectanswerappearsinastandardform.You
mayneedtousestrings,asshowninexamples4and5.
11.8MathQuestionAuthoringusingLaTeX
TheLaTeXauthoringmethodisidealforcreatingsophisticatedmathquestioncontent.Thismethodsupportsthefol-
lowingmathquestiontypes.
DimensionedFormula(MapleFormula(page299))
Numeric(NumericQuestionsinLaTeX(page288))
MultiFormula(MultiFormulaQuestionsin
LaTeX(page296))
Formula(FormulaQuestionsinLaTeX(page291))
Equation(EquationQuestionsinLaTeX(page297))
FormulamodC(ConstantsofIntegrationQuestionsin
LaTeX(page292))
Matrix(MatrixQuestionsinLaTeX(page297))
RestrictedFormula(RestrictedFormulaQuestionsin
LaTeX(page293))
Maple-graded(Maple-gradedQuestionsin
LaTeX(page298))
FormulaList(FormulaListQuestionsin
LaTeX(page294))
InlineMathResponseAreas(MathematicalInline
Questions(page300))
Ntuple(CoordinatesofPointsandVectorsQuestionsin
LaTeX(page294))
OtherMathQuestionFeatures
• Graphplottingapplet-\graphand\multigraphmacros(GraphPlottinginLaTeXQuestions(page279))
• Includinggraphics-\imagemacro(IncludingGraphicReferencesinLaTeXQuestions(page282))
• Labeledgraphics-labelgraphicenvironment(LabelingGraphicsinLaTeXQuestions(page283))
11.8MathQuestionAuthoringusingLaTeX • • 287
SeeAlso:
Overview:AuthoringMethodsforQuestionBanks(page75)
MathQuestionTypesComparisonTable(page153)
NumericQuestionsinLaTeX
QuestionMode:Numeric
TheNumericquestiontypeacceptseitheranumberindecimalorscientificnotation,oranumberwithunitsforan
answer.
Youmustspecifyunitsasanoptionalargumenttothe\answermacro.Forexample:
\answer[m/s^2]{9.8}
Becauseitisnaturaltospecifytheunitsafterthenumber,youcanusetheformat:
\answer{9.8}[m/s^2]
Notes:
• Inthecorrectanswer,thenumbermustbeenclosedinbraces"{}".Theunitsmustbeenclosedinbrackets"[
]".
• Ifyoudonotspecifyunitsinthecorrectanswer(thatis,ifyoudonotspecifyanargumentto\answer),the
systempresentsthestudentwithasingletextfieldinwhichtoenteraresponse(withoutunits).Ifyouspecify
anargumentto\answer,thatargumentistakenasthecorrectanswer'sunits.Inthiscase,thesystemdisplays
twotextfields.Astudentmustenterthenumericalpartoftheresponseinthefirstfieldandtheunitsinthe
secondfield
• Youmustuserecognizedunits.
• Todefineacustomlistofunitsinaquestionbank,usethe\TableOfUnitscommandafterbeginninga
topicintheLaTeXdocument(thatis,after\begin{topic}andbefore\begin{question}{...}).
The\TableOfUnitscommandtakestworequiredarguments,inthefollowingorder:
tablename-tablename,consistingoflower-caseletters(a-z)withnospaces.Usethetablenametoreference
thetableinthedocument.
units_list-thelistofacceptableunits.UnitsaredefinedwithstatementsdefiningtheunitsintermsofSI
unitsorpreviouslydefinedunits.Eachstatementmustendwithasemicolon.Forexample:
cm = 0.01m;
mm = 0.1cm;
ThebaseSIunitsare:
m(meter)
kg(kilogram)
s(second)
A(amp)
K(kelvin)
288 • 11AuthoringQuestionsinLaTeX
cd(Candela)
• Touseacustomlistofunitsinaquestion,usethe\units{tablename}commandinthebodyofthe
question,wheretablenameisthetablename(thefirstargumentto\TableOfUnits).
Important:Becausethesystemconvertsbetweensystemsofunitsusinginexactconversionfactors,itisrecommendedthatyou
acceptresponsesinamarginoferrorinquestionswithunits.
• TheNumericquestiontypehasfivedistincttypesofgrading.Descriptionsaregivenbelow,followedbyusage
instructionsandexamples.
1. Exactvalue-thestudentresponsemustmatchthevaluegiveninthe\answerfield.Thereisnoerrortolerance.
Ifyoudonotspecifyanyothercommands,thisbehaviorisused.
2. Exactvalueandnumberofsignificantdigits-thestudentresponsemustagreewiththevaluegiveninthe\answer
fieldandcontainthespecifiednumberofsignificantdigits.Specifythenumberofsignificantdigitsusingthe
\digits{n}command,wherenisthenumberofsignificantdigits.Forexample,using\digits{3}with
\answer{2.5},only2.50isgradedcorrect.
3. Absolutetolerance-thestudentresponsemustagreewiththevaluegiveninthe\answerfieldtowithinaspecified
tolerance.Specifytheabsolutetoleranceusingthe\err{error}command,whereerroristheabsolutetolerance.
Forexample,using\err{0.25}with\answer{2.5},anyresponsebetween2.25and2.75(inclusive)is
gradedcorrect.Forexample,2.333333iscorrect.
4. Significantdigitswithtolerance-thestudentresponsemustagreewiththevaluegiveninthe\answerfieldto
withinaspecifiedtoleranceinthenthsignificantdigit.Specifythetoleranceusingthe\err{k}and\digits{n}
commands,wherekisthetoleranceinthenthsignificantdigit.Forexample,using\err{2}and\digits{3}
(toleranceof2atthethirdsignificantdigit)with\answer{2.5},anyresponseamong2.48,2.49,2.50,
2.51,and2.52(inclusive)isgradedcorrect.
5. Relativetolerance-thestudentresponsemustagreewiththevaluegiveninthe\answerfieldtowithinaspecified
percenterror.Specifythepercenttoleranceusingthe\perc{error}command,whereerroristhepercent.
Forexample,using\perc{5}with\answer{2.5},anyresponsebetween2.5 * 0.95and2.5 * 1.05
(inclusive)isgradedcorrect.Forexample,2.388888iscorrect.
• Considerthefollowingrecommendationswhenusingthisquestiontype.
1. Ingeneral,allowsometoleranceinastudentresponse.
2. Provideacorrectanswerthatisasaccurateaspossible.
3. Iftheuseofunitsisnotanimportantpartofthequestion(thatis,youarenottestingthestudent'sabilitytoconvert
units),donotincludeunitsinthe\answerfield.Inthiscase,instructthestudentstoenterapurelynumericresponse
intermsofaunityouspecify.
Example1
\begin{question}{numeric}
\qutext{Compute the e exact t value e of the function $f(x)=3x^4-2x^2-1$
at
$x=2.1$.}
\answer{48.5243} % exact value e grading
\end{question}
Example2
\begin{question}{numeric}
\qutext{Compute the e value e of f $\log_{10}(45)$. . Round d your answer to 5
significant digits.}
\answer{1.6532} % % This question uses exact grading g because
\digits is s not t used
% In this case, , a a more accurate response e is s graded d incorrect.
\end{question}
11.8MathQuestionAuthoringusingLaTeX • • 289
Example3
\begin{question}{numeric}
\qutext{Compute the e value e of f $\log_{10}(45)$. . Your answer r must be e correct
to at t least t 5 decimal places.}
\answer{1.653212514}
\err{0.00001} % % absolute e tolerance
% Any y numerical response e between 1.653202514 and 1.653222514 4 is
% graded correct.
\end{question}
Example4
\begin{question}{numeric}
% This question n uses algorithmic c variables.
\qutext{Compute the e value e of f $\log_{10}(\var{a})$. . Your answer r must be
correct to o at t least 5 decimal places.}
\answer{\var{ans}}
\err{0.00001}
\code{
$a=rand(1,100,4);
$ans=log($a);
}
% The e above e code chooses a random value a between 1 and
100, to four
% significant figures. The value of f log(a) ) is s then calculated.
\end{question}
Example5
\begin{question}{numeric}
\qutext{A man is on the top of f a a tower that is 800 0 feet above e ground.
How far can he see?
\newline\newline
Use 3960 miles for the radius of the earth.\newline
Express your answer to an n accuracy y of at t least t 6 significant t digits.}
\answer{34.64134750325308}[mi]
\err{0.0001} % Note: : Using g \digits{6} would allow no tolerance, and would
%
cause '182906 ft' and '34.6413 mi' ' to o be graded incorrect
%
because the values do not match the correct answer.
\end{question}
Example6
\begin{question}{numeric}
% If the ability to convert units is not t an n important part of the
% question, do not use units.
\qutext{A cat is on the top of f a a tower that is 800 0 feet above e ground.
How far (in miles) can it t see?
\newline\newline
Use 3960 miles for the radius of the earth.\newline
Express your answer to an n accuracy y of at t least t 4 decimal places.\newline
Do NOT include any units in your answer.}
\answer{34.64134750325308}
\err{0.0001}
\end{question}
Example7
\begin{question}{numeric}
% An algorithmic version of the previous s question.
% The e height t is s a a multiple of 50 in n [400,900].
\code{$height = = int(range(400,900,50));
$answer
= sqrt((3960 + $height/5280)^2 2 - - 3960^2);}
\qutext{A woman n is s on the e top p of a tower r that is s \var{height}
feet above
ground. How far r (in n miles) can n she e see?
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