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325 lines
12 KiB
HTML
325 lines
12 KiB
HTML
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<title>Activity: Buffon's Needle</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: Buffon's Needle</h1>
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<p class="center">How to estimate <a href="../numbers/pi.html">Pi</a> by dropping a match.</p>
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<div class="center80">
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<p>A few hundred years ago people enjoyed betting on <a href="coin-grid.html">coins tossed on to the floor</a>: would the coin cross a line or not?</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/coin-1750.jpg" alt="coin 1750" height="133" width="240"></p>
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<p>A man (Georges-Louis Leclerc, the <i>Count of Buffon</i>) started thinking about this and worked out the <a href="../data/probability.html">probability</a>.</p>
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<p>It is called "Buffon's Needle" in his honor.</p>
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<p>Now it is your turn to have a go!</p>
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</div>
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<p>You will need:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;"><img src="images/match-no-head.svg" alt="match no head" height="59" width="54"></td>
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<td>
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<p>A <b>match</b>, with the head cut off.<br>
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It must be less than 50 mm.</p>
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<p>(You can use a needle, but be careful!)</p></td>
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</tr>
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<tr>
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<td style="text-align:center;"><a href="images/buffons-needle-grid.gif"><img src="images/buffons-needle-grid.gif" alt="buffons needle grid" height="770" width="888"></a><br>
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</td>
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<td>
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<p>A sheet of paper with lines 50 mm apart.</p>
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<p><input onclick="printImg('images/buffons-needle-grid.gif')" value="Print this one" type="button"></p></td>
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</tr>
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</tbody></table>
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</div>
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<h2>Steps</h2>
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<div class="bigul">
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<ul>
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<li>Measure the spacing of your lines (it may not print at exactly 50mm): ____ mm</li>
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<li>Measure the length of your match (must be less than the line spacing): ____ mm</li>
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<li>Make sure your sheet of paper is on a flat surface such as a table top or the floor.</li>
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<li>From a height of about 5cm, drop the match onto the paper and record whether it lands:
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>
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<p><span class="large">A:</span> Not touching a line</p>
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<p><span class="large">B:</span> Touching or crossing a line</p></td>
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<td><img src="images/buffons-needle-1.gif" alt="buffons needle A B" height="83" width="180"></td>
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</tr>
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</tbody></table></li>
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</ul>
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</div>
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<p>The exact height from which you drop the match is not important, but don't drop it so close to the paper that you are cheating!</p>
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<p>If the match rolls completely off the paper, then do not count that turn.</p>
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<h2>100 Times</h2>
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<p>Now we will drop the match 100 times, but first ...</p>
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<div class="center80">
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<p class="center larger">... what percentage do you think will land A, or B?</p>
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</div>
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<p>Make a guess (estimate) before you begin the experiment:</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<td align="right" height="25">Your Guess for "A" (%):</td>
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<td style="width:80px;"> </td>
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</tr>
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<tr>
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<td align="right" height="25">Your Guess for "B" (%):</td>
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<td> </td>
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</tr>
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</tbody></table>
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<p><b>OK let's begin</b>.</p>
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<p>Drop the match 100 times and record <span class="large">A</span> (does not touch a grid line) or <span class="large">B</span> (touches or crosses a grid line) using <a href="../data/tally-marks.html">Tally Marks</a>:</p>
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<table width="100%" border="1">
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<tbody>
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<tr>
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<td style="text-align:center;">match lands</td>
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<td width="50%">Tally</td>
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<td style="text-align:center;">Frequency</td>
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<td style="text-align:center;">Percentage</td>
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</tr>
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<tr>
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<td style="text-align:center;">
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<div class="large">A</div>(no touch)</td>
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<td height="70"> </td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:center;">
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<div class="large">B</div>(crosses)</td>
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<td height="70"> </td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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<td style="text-align:right;">Totals:</td>
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<td style="text-align:center;">100</td>
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<td style="text-align:center;">100% </td>
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</tr>
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</tbody></table>
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<p>Now draw a <a href="../data/bar-graphs.html">Bar Graph</a> to illustrate your results. You can create one at <a href="../data/data-graph.html">Data Graphs (Bar, Line and Pie)</a>.</p>
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<ul>
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<li>Are the bars the same height?</li>
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<li>Did you expect them to be?</li>
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<li>How does the result compare with your guess?</li>
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</ul>
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<h2>Now Let's Estimate Pi</h2>
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<p>Buffon used the results from his experiment with a needle to estimate the value of <span class="times">π</span> (<a href="../numbers/pi.html">Pi</a>). He worked out this formula:</p>
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<p class="center large"><span class="times">π</span> ≈ <span class="intbl"><em>2L</em><strong>xp</strong></span></p>
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<p>Where</p>
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<ul>
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<li>L is the length of the needle (or match in our case)</li>
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<li>x is the line spacing (50 mm for us)</li>
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<li>p is the proportion of needles crossing a line (case B)</li>
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</ul>
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<p>We can do it too!</p>
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<div class="example">
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<h3>Example: Sam had a match of length 31 mm, and a 40 mm line spacing and 49 of 100 drops crossed the line</h3>
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<p>So Sam had:</p>
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<ul>
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<li>L = 31</li>
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<li>x = 40</li>
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<li>p = 49/100 = 0.49</li>
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</ul>
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<p>Substituting these values into the formula, Sam got:</p>
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<p class="center large"><span class="times">π</span> ≈ <span class="intbl"><em>2 × 31</em><strong>40 × 0.49</strong></span> ≈ 3.16</p>
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</div>
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<p>Now it's your turn. Fill in the following table using <b>your own</b> results:</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<td align="right" height="25">Length of match "<b>L</b>" (mm):</td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25">Line Spacing "<b>x</b>" (mm):</td>
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<td style="width:80px;"> </td>
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</tr>
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<tr>
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<td align="right" height="25"> </td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25"><b>p</b> (the proportion of needles crossing a line):</td>
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<td> </td>
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</tr>
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</tbody></table>
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<p>And do the calculation:</p>
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<p class="center large"><span class="times">π</span> ≈ <span class="intbl">
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<em>2L</em>
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<strong>xp</strong>
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</span> ≈ <span class="intbl">
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<em>2 × _____</em>
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<strong>_____ × _____</strong>
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</span> ≈ _____</p>
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<p>Did you do any better?</p>
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<p>It won't be exact (because it is a random thing) but it may be close.</p>
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<h2>Changing The Subject</h2>
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<p>The next part of this activity is to "<a href="../algebra/equation-formula.html">change the subject</a>" of the formula to work out the perfect value of "p" (the proportion of times the match crosses the line):</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="large"><span class="times">π</span> ≈ 2L/xp</span></span></div>
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<div class="row"><span class="left">multiply both sides by p:</span><span class="right"><span class="large"><span class="times">π</span>p</span> ≈ <span class="large">2L/x</span></span></div>
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<div class="row"><span class="left">divide both sides by <span class="times">π</span>:</span><span class="right"><span class="large">p </span>≈ <span class="large">2L/<span class="times">π</span>x</span></span></div>
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</div>
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<p>And we get:</p>
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<p class="center large">p ≈ <span class="intbl">
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<em>2L</em>
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<strong><span class="times">π</span>x</strong>
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</span></p>
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<div class="example">
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<h3>Example: Alex had a match of length 36 mm, and a 50 mm line spacing.</h3>
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<p>So Alex had:</p>
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<ul>
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<li>L = 36</li>
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<li>x = 50</li>
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</ul>
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<p>Substituting these values into the formula, Alex got:</p>
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<p class="center large">p ≈ <span class="intbl">
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<em>2 × 36</em>
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<strong><span class="times">π</span> × 50</strong>
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</span> ≈ 0.46...</p>
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<p>So Alex should expect the match to cross the line (case B) 46 times out of 100</p>
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</div>
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<p>Fill in the following table using <b>your own</b> results:</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<td align="right" height="25">Length of match "L" (mm):</td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25">Line Spacing "x" (mm):</td>
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<td style="width:80px;"> </td>
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</tr>
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<tr>
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<td align="right" height="25"> </td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25">Estimate for <b>p</b> (≈ 2L/<span class="times">π</span>x):</td>
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<td> </td>
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</tr>
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</tbody></table>
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<p>How close were you?</p>
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<h2>Different Size of Match</h2>
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<p>Try repeating the experiment using a different sized match (but not larger then the line spacing!)</p>
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<ul>
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<li>Did you get better or worse results?</li>
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</ul>
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<h2>What You Have Done</h2>
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<p>You have (hopefully) had fun running <b>an experiment</b>.</p>
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<p>You have had some experience with calculations.</p>
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<p>And you have seen the relationship between theory and reality.</p>
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<p> </p>
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<div class="related">
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<a href="coin-grid.html">Activity: Dropping a Coin onto a Grid</a>
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<a href="pi-approximation.html">Activity: Find an Approximate Value For Pi</a>
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</div>
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<!-- #EndEditable -->
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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