<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>18.4.4.5 Exercise 4</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>If \(\displaystyle{n=\frac{1}{2L}\sqrt{\frac{E}{d}}}\) and \(L, E\) and \(d\) can be measured correct to within \(1\)%, how accurate is the calculated value of \(n\)?
</p>
<p>The maximum percentage error in \(n\) is [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]%.</p>
<p></p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>We are given the funtcion \(\displaystyle{n=\frac{1}{2L}\sqrt{\frac{E}{d}}}\) and \(L, E, d\) are subject to percentage relative errors of \({@error_L@} \%\).</p>
<p>First, find \(\dfrac{\partial n}{\partial L},\) \(\dfrac{\partial n}{\partial E}\) and \(\dfrac{\partial n}{\partial d}\)</p>
<p>\(\dfrac{\partial n}{\partial L} = \displaystyle {@n_L@}\)</p>
<p>\(\dfrac{\partial n}{\partial E} = \displaystyle {@n_E@}\)</p>
<p>\(\dfrac{\partial n}{\partial d} = \displaystyle {@n_d@}\)<br></p>
<p>Now we write down the expression of \( \delta n .\)</p>
<p>\[ \begin{align} \left|\delta n \right|_{\text{max}} &amp; \simeq \left| \frac{\partial n}{\partial L} \right| \left| \delta L \right|+ \left| \frac{\partial n}{\partial E} \right| \left| \delta E \right| + \left| \frac{\partial n}{\partial d} \right| \left|
    \delta d \right|&nbsp;\\ &amp; \simeq \left| {@n_L@} \right| \left| \delta L \right| + \left| {@n_E@} \right| \left| \delta E \right| + \left| {@n_d@} \right| \left| \delta d \right| \end{align}\]</p>
<p>Hence we write down an expression for the percentage relative error in \(f\):</p>
<p>\[ \begin{align} \dfrac{\delta n}{n} \times 100 &amp; \simeq \left| {@n_L@} \times 2L \sqrt{\frac{d}{E}} \, \delta L \times 100 \right| + \left| {@n_E@} \times 2L \sqrt{\frac{d}{E}} \, \delta E \times 100 \right| +&nbsp;\left| {@n_d@} \times 2L \sqrt{\frac{d}{E}}
    \, \delta d \times 100 \right| \\ &amp; \simeq \left| {@n1@} \delta L \times 100 \right| + \left| {@n2@} \delta E \times 100 \right| +&nbsp;\left| {@n3@} \delta d \times 100 \right| \end{align}\]</p>
<p>The percentage relative errors of \(L,E, d\) are \({@error_L@} \%\) so:</p>
<p>\[ \frac{\delta L}{L} = {@error_L@}, \quad \frac{\delta E}{E} = {@error_E@}, \quad \frac{\delta d}{d} = {@error_d@} \]</p>Substituting them into the previous relation
we find:<br>\[\dfrac{\delta v}{v} \times 100 \simeq&nbsp; {@error_L@} +\frac{{@error_E@}}{2} +&nbsp;\frac{{@error_d@}}{2} ={@ta1@} \%\]<br>
<p></p>
<p><span style="font-size: 0.9375rem;"><strong>Alterantive method:</strong></span></p>
<p>Take logarithms of both sides \[\ln n = -\ln 2 - \ln L + \frac{1}{2} \ln E - \frac{1}{2} \ln d\]</p>
<p>So \(\displaystyle \frac{\delta n}{n} = -\frac{\delta L}{L}+ \frac{\delta E}{2E} - \frac{\delta d}{2d}\)</p>
<p>The maximum percentage error in \(n\) is therefore \[v=\left|\frac{\delta L}{L}\right|+\left|-\frac{\delta E}{2E}\right| + \left|-\frac{\delta d}{2d}\right|=&nbsp;{@error_L@} +\frac{{@error_E@}}{2} +&nbsp;\frac{{@error_d@}}{2} = {@ta1@} \% \]</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text>n: 1/(2*L)*sqrt(E/d);
n_inv: 1/n;

n_L: diff(n,L);
n_E: diff(n,E);
n_d: diff(n,d);

n1: n_inv*n_L;
n2: n_inv*n_E;
n3: n_inv*n_d;

error_L: 1;
error_E: 1;
error_d: 1;

ta1: (error_L)+(error_E/2)+(error_d/2);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>{@a@}</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>numerical</type>
      <tans>ta1</tans>
      <boxsize>5</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
      <forbidwords/>
      <allowwords/>
      <forbidfloat>0</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>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ta1</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>prt1-1-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ev(ta1+1,simp)</value>
      </testinput>
      <expected>
        <name>prt1</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>prt1-1-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
