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<quiz>
  <question type="stack">
    <name>
      <text>14.3.1.2 Find volume of cone</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the volume of the cone generated by rotating \(y={@y@}\), for \({@low@}\le x\le {@upp@}\), around the \(x\)-axis. In order to find the volume of this solid we assume that it is composed of lots of thin circular discs all aligned perpendicular to
    the \(x\)-axis, such as that shown in the diagram. From the diagram below we note that a typical disc has radius \(y\), which in this case equals \({@y@}\), and thickness \(\delta x\).</p>
<p style="text-align: center;"><img src="@@PLUGINFILE@@/14-3-%28modified%29diagram-conedisk.svg" alt="Representation of a circular disk on the cone of revolution." width="287" height="361" class="img-responsive atto_image_button_text-bottom"><br></p>

<p>The cone is divided into a number of thin circular discs.</p>
<p>The volume of a circular disc is the circular area multiplied by the thickness.</p><br>
<p>Write down an expression for the volume of this typical disc. You can type \(\delta x\) as <code>deltax</code>.</p>
<p>\(\quad V_{\mbox{disc}}=\) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p><br>

<p>To find the total volume we must sum the contributions from all discs and find the limit of this sum as the number of discs tends to infinity and \(\delta x\) tends to zero. That is \[ \lim_{\delta x\rightarrow 0} \sum_{x={@low@}}^{x={@upp@}} {@areaC@}
    \delta x\] This is the definition of a definite integral.</p><br>
<p> Write down the corresponding integral:</p>
<table>
    <tbody>
        <tr>
            <td>\(\displaystyle \int\)</td>
            <td>
                <table>
                    <tbody>
                        <tr>
                            <td>[[input:ans2up]] [[validation:ans2up]]</td>
                        </tr>
                        <tr>
                            <td>[[input:ans2dn]] [[validation:ans2dn]]</td>
                        </tr>
                    </tbody>
                </table>
            </td>
            <td> [[input:ans2int]] [[validation:ans2int]] \(\mathrm{d}x\)</td>
        </tr>
    </tbody>
</table>
<p>[[feedback:prt2]]</p><br>

<p>Find the required volume by performing the integration:</p>
<p>\(\quad V =\) [[input:ans3]] [[validation:ans3]] [[feedback:prt3]]</p>]]></text>
<file name="14-3-(modified)diagram-conedisk.svg" path="/" encoding="base64">[encoded data, 33288 characters]</file>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<h4 class="HELM_solution">Solution</h4>
<p>A disc is like a cylinder. So the volume of the disc will be given by the area of the circle 'base', which has radius \({@y@}\), multiplied by the thickness of the disc, \(\delta x\), so that \[V_{\mbox{disc}} = {@ta1@}.\]</p>
<p>The integral corresponding to the limit given is \[\int_{@ta2dn@}^{@ta2up@} {@ta2int@} \,\mathrm{d}x.\]</p>
<p>Evaluating that integral, we obtain the volume of the cone: \[V={@ta3@}.\]</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text><![CDATA[texput(deltax,"\delta x");
y: rand([2,3,5])*x;
low: 0;
upp: rand([2,3,4]);

areaC: simplify(y^2*pi);
ta1: areaC*deltax;

ta2up: upp;
ta2dn: low;
ta2int: areaC;

ta3: simplify(integrate(ta2int,x,ta2dn,ta2up));]]></text>
    </questionvariables>
    <specificfeedback format="html">
      <text/>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\(y=\) {@y@}; \({@low@}\le x\le {@upp@}, \, V= {@ta3@}\)</text>
    </questionnote>
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      <text/>
    </questiondescription>
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    <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> Your answer is partially correct.</span>]]></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>
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