<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>12.2.2.5 maximum cross-sectional area</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>A materials store is to be constructed next to a \(3\) metre high stone wall (shown as \(OA\) in the cross section in the diagram). The roof \((AB)\) and front \((BC)\) are to be constructed from corrugated metal sheeting. Only {@const@} metre length
    sheets are available. Each of them is to be cut into two parts such that one part is used for the roof and the other is used for the front. Find the dimensions \(x, y\) of the store that result in the maximum cross-sectional area. Hence determine
    the maximum cross-sectional area.
</p>
<figure class="HELM_figure"><img src="@@PLUGINFILE@@/12.2.7%20stone%20wall.svg" alt="stone wall to left 3 metres, top left A top right B y right length, x bottom length, O bottom left and C bottom right." class="img-responsive atto_image_button_text-bottom" width="171" height="156"></figure>

<ol class="HELM_parts">
    <li>
        Give the cross sectional area of the store, \(A=\) [[input:ans1]] [[validation:ans1]] <br>[[feedback:prt1]]
    </li>

    <li>
        Give the equation that relates \(AB\) and \(BC\): [[input:ans2]] [[validation:ans2]] (use <code>AB</code> and <code>BC</code> as variables)<br>[[feedback:prt2]]
    </li>

    <li>
        Now, rewrite it as a polynomial in terms of \(x\) and \(y\): [[input:ans3]] [[validation:ans3]] <br>[[feedback:prt3]]
    </li>

    <li>
        Calculate the values of \(x\) and \(y\) that maximise the cross sectional area (give a precise answer):<br> \(\quad x=\) [[input:ans4]] [[validation:ans4]] <br> \(\quad y=\) [[input:ans5]] [[validation:ans5]] <br> [[feedback:prt4]]
    </li>

    <li>
        Finally, give the maximum value of the cross sectional area (give a precise answer): \(\max(A) =\) [[input:ans6]] [[validation:ans6]] <br> [[feedback:prt5]]
    </li>
</ol>

(The following information pages are applications, if you want to move to the next exercise simply move to 6 in the menu on the left)]]></text>
<file name="12.2.7 stone wall.svg" path="/" encoding="base64">[encoded data, 254120 characters]</file>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<ol class="HELM_parts">

<li><p dir="ltr" style="text-align: left;">Note that the store has the shape of a trapezium. So the cross-sectional area (\(A\)) of the store is given by the formula: Area \(=\) average length of parallel sides \(\times\) distance between parallel sides:<br>\[<br>A=\frac{1}{2}(y+3)x \qquad\qquad
    (1)
    <br>\]</p></li>

<li><p>The lengths \(x\) and \(y\) are related through the fact that \(AB+BC= {@const@}\), where \(BC=y\) and \(AB=\sqrt{x^2+ (3-y)^2}\). </p></li>

<li><p>Hence\(\sqrt{x^2+ (3-y)^2}+y= {@const@}\). This equation can be rearranged in the following way:<br>\[<br>\sqrt{x^2+
    (3-y)^2}= {@const@}-y \iff x^2+ (3-y)^2= ({@const@}-y)^2\quad \mbox{ i.e. }\;\; {@expand(x^2+(3-y)^2)@}= {@expand((const-y)^2)@}<br>\]<br>which implies that \[{@ta3@}.\qquad\qquad (2)\]</p></li>

<li><p>It is necessary to eliminate either \(x\) or \(y\) from (1) and (2) to obtain an equation in a single variable. Using \(y\) instead of \(x\) as the variable will avoid having square roots appearing in the expression for the cross-sectional area. Hence<br>\[<br>{@eq4@}.\qquad
    \qquad (3)<br>\]<br>Substituting for \(y\) from Equation (3) into Equation (1) gives <br>\[<br>A={@ev(cross_area,eq4)@}={@area_sub@}<br>\]<br>To find turning points, we evaluate \(\dfrac{\mathrm{d}A}{\mathrm{d}x}\) from above to get <br>\[<br>\dfrac{\mathrm{d}A}{\mathrm{d}x}={@darea@}<br>\]<br>Solving
    the equation \(\dfrac{\mathrm{d}A}{\mathrm{d}x}= 0\) gives \(\displaystyle {@darea@} = 0 \;\implies\; {@simplify(darea=0)@}\).
</p>
<p dir="ltr" style="text-align: left;"> Hence \(x=\pm {@ta4@}=\pm {@x_approx@}\). Only \(x&gt;0\) is of interest, so \(x={@ta4@} = {@x_approx@}\) gives the required turning point. <br></p>
<p dir="ltr" style="text-align: left;"><strong>Check</strong>: Differentiating \(\dfrac{\mathrm{d}A}{\mathrm{d}x}\) and using the positive \(x\) solution gives <br>\[<br>\frac{\mathrm{d}^2A}{\mathrm{d}x^2}={@diff(darea,x)@}={@subst(point[1],diff(darea,x))@} &lt;0<br>\]<br>Since the second derivative is negative then the cross-sectional
    area is a maximum. This is the only turning point identified for \(A &gt;0\) and it is identified as a maximum. To find the corresponding value of \(y\), substitute \(x= {@ta4@}={@x_approx@}\) into \({@eq3@}\) to get \( {@ev(eq3,point[1])@}= {@y_approx@}\).
    So the values of \(x\) and \(y\) that yield the maximum cross-sectional area are \({@x_approx@}\,m\) and \({@y_approx@}\,m\) respectively. </p> </li>

<li><p> To find the maximum cross-sectional area, substitute for \(x={@x_approx@}\) into \(\dfrac{\mathrm{d}A}{\mathrm{d}x}\) to get \(A={@max_area@}
    = {@A_approx@}\). So the maximum cross-sectional area of the store is \({@decimalplaces(A_approx,2)@}\,m^2\)to \(2\) d.p.<br></p></li>
</ol>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text><![CDATA[/*Randomisation constants*/
const: rand([5,6,7,8]);

/*Question variables*/

cross_area: 1/2*(y+3)*x;
metal_eqtn: AB+BC = const;
roof: sqrt(x^2+(3-y)^2);
front: y;

eq0:subst(roof,AB,metal_eqtn);
eq1: subst(front,BC,eq0);
eq2: roof^2=(const-y)^2;
eq3: solve(eq2,x^2)[1];
eq4: solve(eq2,y)[1];

area_sub: expandwrt(ev(cross_area,eq4),x^2)
darea: diff(area_sub, x);
points: solve([eq4,darea=0],[x,y]);
point: sublist(points, lambda([x], is(rhs(x[1])>0)))[1];
max_area: subst(point,cross_area);


/*Answers*/
ta1: cross_area;
ta2: metal_eqtn;
ta3: eq3;
ta4: rhs(point[1]);
ta5: rhs(point[2]);
ta6: max_area;

/*Display*/
x_approx: decimalplaces(float(ta4),4);
y_approx: decimalplaces(float(ta5),4);
A_approx: decimalplaces(float(max_area),4);]]></text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\[ l = {@const@}, A= {@ta1@}, \quad {@ta3@}\]
\[ x= {@ta4@}, y={@ta5@}, max(A)= {@ta6@} \]</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>ta1</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>0</requirelowestterms>
      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans2</name>
      <type>algebraic</type>
      <tans>ta2</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>0</requirelowestterms>
      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans3</name>
      <type>algebraic</type>
      <tans>ta3</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>0</requirelowestterms>
      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans4</name>
      <type>algebraic</type>
      <tans>ta4</tans>
      <boxsize>5</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>0</requirelowestterms>
      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans5</name>
      <type>algebraic</type>
      <tans>ta5</tans>
      <boxsize>5</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>0</requirelowestterms>
      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <input>
      <name>ans6</name>
      <type>algebraic</type>
      <tans>ta6</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>1</forbidfloat>
      <requirelowestterms>0</requirelowestterms>
      <checkanswertype>0</checkanswertype>
      <mustverify>1</mustverify>
      <showvalidation>3</showvalidation>
      <options/>
    </input>
    <prt>
      <name>prt1</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>ta1</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt1-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt1-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt2</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans2</sans>
        <tans>ta2</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt2-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt2-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt3</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text>sa3: if equationp(ans3) then lhs(ans3)-rhs(ans3) else ans3;
sa4:polynomialpsimp(sa3);</text>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>equationp(ans3)</sans>
        <tans>true</tans>
        <testoptions/>
        <quiet>1</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty/>
        <truenextnode>1</truenextnode>
        <trueanswernote>prt3-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt3-1-F</falseanswernote>
        <falsefeedback format="html">
          <text>Your answer must be an equation!</text>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>sa4</sans>
        <tans>true</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.0000000</truescore>
        <truepenalty/>
        <truenextnode>2</truenextnode>
        <trueanswernote>prt3-2-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt3-2-F</falseanswernote>
        <falsefeedback format="html">
          <text>Your answer must be a polynomial!</text>
        </falsefeedback>
      </node>
      <node>
        <name>2</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>sa3=0</sans>
        <tans>ta3</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt3-3-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt3-3-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt4</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans4</sans>
        <tans>ta4</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>0.5000000</truescore>
        <truepenalty/>
        <truenextnode>1</truenextnode>
        <trueanswernote>prt4-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>1</falsenextnode>
        <falseanswernote>prt4-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
      <node>
        <name>1</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans5</sans>
        <tans>ta5</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>+</truescoremode>
        <truescore>0.5000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt4-2-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>-</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt4-2-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <prt>
      <name>prt5</name>
      <value>1.0000000</value>
      <autosimplify>1</autosimplify>
      <feedbackstyle>1</feedbackstyle>
      <feedbackvariables>
        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans6</sans>
        <tans>ta6</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>prt5-1-T</trueanswernote>
        <truefeedback format="html">
          <text/>
        </truefeedback>
        <falsescoremode>=</falsescoremode>
        <falsescore>0.0000000</falsescore>
        <falsepenalty/>
        <falsenextnode>-1</falsenextnode>
        <falseanswernote>prt5-1-F</falseanswernote>
        <falsefeedback format="html">
          <text/>
        </falsefeedback>
      </node>
    </prt>
    <deployedseed>1201096321</deployedseed>
    <deployedseed>120718477</deployedseed>
    <deployedseed>1685978888</deployedseed>
    <deployedseed>53549973</deployedseed>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>ta2</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>ta3</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value>ta6</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt2-1-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-3-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt4-2-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt5-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1+1</value>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value>rhs(ta2)</value>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>rhs(ta3)</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value>ta4+1</value>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value>ta5+1</value>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value>ta6+1</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt2-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt3-1-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt4-2-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt5-1-F</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>3</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>eq2</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value/>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-3-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>4</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>eq4</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value/>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt3-3-T</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>5</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans2</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans3</name>
        <value>sqrt(x^2+ (3-y)^2)= const-y</value>
      </testinput>
      <testinput>
        <name>ans4</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans5</name>
        <value/>
      </testinput>
      <testinput>
        <name>ans6</name>
        <value/>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt2</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt3</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty/>
        <expectedanswernote>prt3-2-F</expectedanswernote>
      </expected>
      <expected>
        <name>prt4</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
      <expected>
        <name>prt5</name>
        <expectedscore/>
        <expectedpenalty/>
        <expectedanswernote>NULL</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
