7359
/www/nrich/html/content/id/7359/
1
2011-03-30T13:58:58
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<mdoxml version="1.0"><ul id="stemLinks">
<li><a href="http://nrich.maths.org/7322">Warm-up</a></li>
<li><a href="http://nrich.maths.org/6683">Try this next</a></li>
<li><a href="http://nrich.maths.org/7109">Think higher</a></li>
<li><a href="http://nrich.maths.org/1477">Read: mathematics</a></li>
<li><a href="http://plus.maths.org/content/leaning-2012">Read: technology</a></li>
<li><a href="http://en.wikipedia.org/wiki/Olympic_Stadium_(London)">Explore further</a></li>
</ul>
<div> </div>
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<p>Imagine you are building a new Olympic stadium and you are responsible for designing and marking out the running track. The track needs to fulfil the following specifications:<br></br>
</p>
<ul>
<li>The distance around the inside edge of the inner lane should be 400m.</li>
<li>There should be 8 lanes. </li>
<li>Each lane should be 1.25m wide.</li>
<li>The track should consist of two straight sections joined by two semi-circular sections. </li>
<li>The straight sections should each be 85m in length (a straight section is extended over the curve for the 100m race, as shown below). </li>
</ul>
<br></br>
<div style="text-align: center;"><mdo:image alt="diagram of track" height="142" src="track.jpg" width="362"></mdo:image></div>
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<p> <br></br>
<span style="font-weight: bold;">Can you work out the radius of the curved sections in order to produce an accurate scale drawing?</span><br></br>
<br></br>
For the 200m race, runners start on the curved section at the right of the diagram and run anticlockwise to the finish line at the top left.<br></br>
As the outer lanes are longer than the inner lanes, a staggered start is needed so that at the finish line all runners have run the same distance.<br></br>
<br></br>
<span style="font-weight: bold;">Can you work out where each runner should start so that they all run 200m?</span><br></br>
<br></br>
For the 400m race, the runner in lane 1 does one complete lap of the track, so the start line is the same as the finish line. The runners in lanes 2 to 8 again have a staggered start.<br></br>
<br></br>
<span style="font-weight: bold;">Can you work out where each runner should start so that they all run 400m?</span><br></br>
<br></br>
<br></br>
</p>
<div class="notes"> </div>
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<span class="editorial">Well done to Daniel from Kings School, New
Zealand, Josh from Chatham Grammar School for Boys, and Anna
from the Sandon School, who all correctly worked out the dimensions
of the running track. Here is Josh's solution:</span><br></br>
<br></br>
Radius of the curved sections:<br></br>
<br></br>
As the straight sections of the track are both 85m long, 170m of
the track is straight. That means that the remaining 230m of the
track is comprised of the two semicircular sections. As there are
two semicircular sections, each of equal radii, the sum of their
inside edges (of the inside lanes) is equal to the circumference of
a circle that would fit neatly into the inside curves of the
track.<br></br>
<br></br>
The circumference of a circle, $c$, is given by the equation: $c =
2 \times \pi \times r$, where $r$ is the radius.<br></br>
<br></br>
Therefore, as we have deduced that the circumference of this
imaginary circle is equal to the sum of the inside curves of the
track, we can write: $230 = 2 \times \pi \times r$, so $r =
\frac{230}{2\pi} = 36.606$ metres (to 3 d.p.)<br></br>
<br></br>
This measurement is only the measurement of the radii of the
semicircles which comprise the inside edge of the inside lane on
the curved section. We want to find a measurement for the radii of
the semicircles which comprise the whole of both curved sections;
there are 8 lanes that we must consider in order to do this. As
each lane is a constant 1.25 metres, the sum of the thickness of
the lanes is equal to 8 x 1.25 = 10 metres. Therefore, the radius
of the curved sections, in order to produce a scale drawing, is
equal to: 36.606 + 10 = 46.606 metres (to 3 d.p.)<br></br>
<br></br>
The 200m Staggered Start:<br></br>
<br></br>
As it is mentioned in the problem for the 400m start that the
measurement for the length of a given lane is that of its inside
edge, I will assume that it is the inside edges of each lane that
must be made constant in the 200m race in order to make the race
fair. The runner in lane 1 starts at the curved section on the
bottom right of the track, so I will use that lane for comparison
with the others. The straights do not need to be compensated for in
the staggered start, as the runners would all run the same distance
here. Therefore, we only need to consider the 115m of track that
the runners will run on before they hit the straight; in this time,
all runners will be running on the curved section of track.<br></br>
<br></br>
The runner in lane 2 is running on a track with an inside edge that
is 1.25m further outwards than that of lane 1. Therefore, the
radius of runner 2's track is equal to $36.606 + 1.25 = 37.856$
metres. Thus, the circumference of the semicircle that makes up the
curved section is equal to: $\frac{2 \pi r}{2} =
\pi r = 118.927$ metres This is 3.927 metres longer than the
inside edge of lane 1, so the rune in lane 2 will start 3.927
metres in front of the runner in lane 1.<br></br>
<br></br>
We can also generate an nth term sequence for the lanes. The nth
term for this sequence is: $(36.606 + 1.25(n-1)) \times \pi$ I
arrived at this conclusion because the formula for the arc of a
semicircle is pi x r, hence the reason for multiplying the sequence
by pi, and the section in brackets is the way to determine the
radius of the inside edge of any given track. Each runner, as we
move from lane 1 to lane 8, will start 3.927 metres (3 d.p.) in
front of the previous runner.<br></br>
<br></br>
The 400m Staggered Start:<br></br>
<br></br>
The obvious conclusion for this problem would be to say that each
runner starts twice as far behind the runner in the next inside
lane as they did in the 200m race, but I will investigate this
mathematically.<br></br>
<br></br>
Each runner is running 400 metres, but 170 metres of the track
they will run (85 x 2) is made up of straight sections, where the
runners will run the same distance regardless of the lane they are
in, so this does not need to be compensated for. Thus, 230m of the
track they will run on is comprised of a curved section, which must
be compensated for. When the runner in lane 2 reaches a curved
section, the radius of the inner edge of his/her lane is equal to
$36.606 + 1.25 = 37.856$ metres. Thus, the arc lengths of both
semicircles together is equal to: $2 \times \pi \times 37.856
= 237.854$ metres (3 d.p.) This is 7.854 metres longer than the
length of curved track in lane 1, so the runner in lane 2 will
start 7.854 metres in front of the runner in lane 2.<br></br>
<br></br>
We can also generate an nth term sequence for the lanes. The nth
term for this sequence is: $2 \times ((36.606 + 1.25(n-1))
\times pi)$. I arrived at this conclusion using the nth term
formula I generated before to calculate the arc length of one of
the semicircular pieces of track, then multiplied it by 2 (for the
two semicircles involved in the 400m. As we move from lane 1 to
lane 8, each runner starts 7.854 metres in front of the runner on
the next inside lane to them. <br></br>
<br></br>
<span class="editorial">Rajeev, from Haberdashers' Aske's Boys'
School, sent us his calculations for the staggers presented clearly
in a table. You can see it</span> <a href="/content/id/7359/Rajeev%20solution.pdf" class="editorial">here</a><span class="editorial">.</span><br></br>
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<h3>Why do this problem?</h3>
This problem offers an authentic context within which to calculate arc lengths and requires students to present their findings in a convincing manner.<br></br>
<h3>Possible approach</h3>
<div>Arrange the class in groups of three or four, and hand out <a href="/content/id/7359/track%20design.pdf">this worksheet</a> to each group.</div>
<div> </div>
<div>"Your task is to work out the dimensions of a running track that satisfy all the criteria on the worksheet, and to work out where to mark the start lines for the 200m (and/or 400m) race."</div>
<div> </div>
Give the students plenty of time to discuss and work on the problem in their groups. For groups not used to working collaboratively, these <a href="/content/id/7359/Roles.pdf">roles</a> may be useful to guide students in organising themselves to work together.<br></br>
<br></br>
To finish off, results could be presented as a list of the dimensions/angles that a groundskeeper would need in order to paint the lines for the running track, together with explanations of how students worked them out.<br></br>
<br></br>
Additionally, students could be asked to produce a scale drawing of the track design to an agreed scale (1cm to 2.5m would fit on a large flipchart sheet) and then each group's drawing could be overlaid on another group's to check to see if they coincide.<br></br>
<h3>Key questions</h3>
<div>If a straight line section is 85m, what is the length of a curved section, and what is its radius?</div>
<div>If you moved from the first lane to the second lane, how much further would you run on a complete lap of the track? What about in the third lane? Or fourth lane...?</div>
<h3>Possible extension</h3>
<div>In reality, the 400m is measured 30cm in from the inside edge of the inside lane and 20cm in from the inside edges of the other lanes. Students could be given this extra layer of complexity to include in their working. </div>
<div> </div>
<div><a href="http://nrich.maths.org/2095&part=">Triangles and Petals</a> offers students the chance to investigate arc lengths in a different context.</div>
<div> </div>
<div>Some runners might complain that the sharper bend on the inner lane makes it more difficult to run around. Can students design a track which would reduce these effects?</div>
<br></br>
<h3>Possible support</h3>
By working in groups, students are encouraged to support each other in making sense of the problem and working towards a solution.<br></br>
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<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">If you know the radius of a semicircle, how do you find the length
of the arc?<br></br>
If you know the arc length, how do you find the radius?<br></br></mdoxml>
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Track design
Where should runners start the 200m race so that they have all run the same distance by the finish?
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