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243 lines
9.8 KiB
HTML
243 lines
9.8 KiB
HTML
<!DOCTYPE html>
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<!-- Mirrored from www.mathsisfun.com/activity/how-high.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:58:22 GMT -->
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<title>Activity: How High?</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: How High?</h1>
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<p>For this activity you will need:</p>
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<ul>
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<li>A piece of cardboard or thin sheet of wood, about the size
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of a sheet of paper.</li>
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<li>A piece of string about 30cm or 1 foot long:</li>
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</ul>
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<img alt="string" src="images/howhigh1.jpg">
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<ul>
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<li>A drawing pin (thumbtack) or a small nail:</li>
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</ul>
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<img alt="pin" src="images/howhigh2.jpg">
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<ul>
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<li>A small rock (one you can tie the string to):</li>
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</ul>
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<img alt="howhigh3" src="images/howhigh3.jpg">
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<ul>
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<li>A tape measure:</li>
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</ul>
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<img alt="tape measures" src="images/howhigh4.jpg">
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<ul>
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<li>scissors</li>
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<li>glue</li>
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</ul>
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<h2>Aim</h2>
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The aim of this activity is to measure the height of a tall building or
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tree.<br>
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I will illustrate this with a tree, but the method is just the
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same for a building.<br>
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<br>
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<p class="center"><img alt="tree" src="images/howhigh5.jpg"></p>
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<h3> </h3>
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<h3>Suppose you want to measure the height.</h3>
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<p><strong>Can I just measure it with
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a tape measure?</strong><br>
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That won't be easy. You may have to climb the tree to do that. Dangerous!</p>
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<p>But there is another, easier, way!</p>
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<div class="center80">
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<p>Measure a length <b>along the ground</b> and measure the <b>angle θ</b> to the top of the tree:</p>
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<p class="center"><img alt="tree with angle" src="images/howhigh6.jpg"></p>
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<p><i>Surveyors and engineers use special equipment to measure these
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distances and angles accurately. The instrument they use to measure the
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angle is called a theodolite.
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In this activity you will make a simple theodolite. You won't get
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accurate answers, but it will help you to understand how surveyors do
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their work.</i></p>
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</div>
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<p><strong>There are three parts to this activity:</strong></p>
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<ul>
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<li>Making a simple theodolite.</li>
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<li>Measuring the angle and the distance along the ground.</li>
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<li>Using the measurements to find the height of the tree.</li>
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</ul>
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<h2>Making a Simple Theodolite</h2>
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<p><strong>Step 1:</strong> Print
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out this <a href="images/howhigh7.jpg">image</a> of a protractor onto A4 paper.<br>
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You will need to change the page
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orientation to landscape. </p>
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<p><strong>Step 2:</strong> You do not need to cut
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out the picture of the protractor, but you will need to cut as near as
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you can to the top edge - just leave a small space, enough for you to
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attach the drawing pin or nail.</p>
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<p><strong>Step3:</strong> Glue your picture of the
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protractor with its straight-edge lying along the<br>
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top edge of your
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cardboard or timber:</p>
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<p class="center"><img alt="protractor on wood" src="images/howhigh8.svg"></p>
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<p><strong>Step 4:</strong> Tie the small rock to
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one end of the piece of string:</p>
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<p class="center"><img alt="rock on string" src="images/howhigh9.jpg"></p>
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<p><strong>Step 5:</strong> Attach the other end of
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the string to the top middle point of the <br>
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protractor using the drawing
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pin or
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nail:</p>
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<p class="center"><img alt="protractor on wood with rock on string" src="images/howhigh10.jpg"></p>
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<p>When you allow the rock to pull the string downwards, gravity will make
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it hang vertically. Even if you tilt the piece of cardboard or timber
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at an angle, the string will still hang vertically:</p>
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<p class="center"><img alt="protractor string at 90" src="images/howhigh11a.svg"> <img alt="protractor string at 60" src="images/howhigh11b.svg"></p>
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<p><strong>Can you see how the angle of 30°
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is related to the angle shown by the string on the protractor?</strong></p>
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<h2><br>
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Measuring the Angle
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and the Distance Along the Ground</h2>
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<p>First of all, find a good place to stand about <strong>100 feet or 30m
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away from the tree</strong>. Make sure that you are standing on level ground and
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in a safe place. Mark the place with a stick. Now you are ready to
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measure.</p>
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<p>To measure the angle, you just point your 'theodolite'' towards the top
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of the tree. Put your eye as close as possible to one edge of the
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protractor; then, looking along the straight edge, point it directly at
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the top of the tree:</p>
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<p class="center"><img alt="howhigh12" src="images/howhigh12.jpg"></p>
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<p>Hold the string and weight in position before you lower the theodolite.
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Then read the angle from the theodolite as the angle the vertical
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weight on
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the string makes with the protractor. Adjust your answer to find
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the <a href="../algebra/trig-finding-angle-right-triangle.html">angle
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of elevation</a>, <strong>θ</strong>, from
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your eye to the top of the tree.</p>
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<p>To find the distance, <strong>d</strong>, use your tape measure to measure from the
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stick to the base of the tree, or more exactly to the midpoint of the
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base of the tree - <strong>can you see how to make allowance for that?</strong></p>
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<h2>Using the
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Measurements to Find the Height of the Tree</h2>
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<p>You can calculate the height of the tree using a scale diagram on a
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piece of paper,<br>
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following these steps:</p>
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<ul>
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<li>Draw a horizontal line:</li>
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</ul>
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<img alt="howhigh13" src="images/howhigh13.jpg">
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<ul start="2">
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<li>At a point, <strong>C</strong>, near the right end of the line, measure an angle
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equal to the angle, <strong>θ</strong>, from your experiment:</li>
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</ul>
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<img alt="howhigh14" src="images/howhigh14.jpg">
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<ul start="3">
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<li>Choosing a suitable scale, measure the distance x equivalent to
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the distance <strong>d</strong> you measured in your experiment. Mark a point <strong>B</strong>:</li>
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</ul>
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<img alt="howhigh15" src="images/howhigh15.jpg">
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<ul start="4">
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<li>Draw the perpendicular at <strong>B</strong> to meet the sloping line at <strong>A</strong>:</li>
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</ul>
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<img alt="howhigh16" src="images/howhigh16.jpg">
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<ul start="5">
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<li>Measure the height, <strong>y</strong>, of <strong>A</strong> above <strong>B</strong>:</li>
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</ul>
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<img alt="howhigh17" src="images/howhigh17.jpg">
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<ul start="6">
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<li>Using the scale from step 3, convert <strong>y</strong> back to find the height, <strong>h</strong>, of the tree.</li>
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</ul>
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<h2>Wait!</h2>
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<ul class="bigul">
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<li>Did you notice anything wrong?</li>
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<li>Where was your eye when you measured the angle θ?</li>
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<li>Can you think of a way to compensate for that error?</li>
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</ul>
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<h2>Another Way</h2>
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<p>There is an alternative way to calculate the height of the tree using <a href="../sine-cosine-tangent.html">trigonometry</a>.<br>
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If you know how to do this, then go ahead.</p>
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<ul>
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<li>How accurate will your answer be?</li>
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</ul>
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<h3>Can you list below some things about this experiment that tell you that
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your answer is not very accurate?</h3>
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<ol>
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<li> </li>
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<li> </li>
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<li> </li>
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<li> </li>
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<li> </li>
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</ol>
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<h3>Any more?</h3>
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<p><strong>Try to find out how</strong><strong> a real theodolite is made and why it gives much
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more accurate measurements.</strong></p>
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<p> </p>
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<div class="related">
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<a href="index.html">Activities Index</a>
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