6308
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2011-02-01T00:00:01
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<p>Scientific measurement allows you to practise your measurement skills in a variety of intriguing biological settings.<br></br>
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<mdo:flash height="600" id="/content/id/6308/measures.swf" width="800"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
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<a href="/content/id/6308/measures.swf">Click here</a> to view in full screen mode.<br></br>
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<h6>Send <a href="mailto:post16.nrich@maths.org?subject=Feedback%20on%20NRICH%20problem%20id%206308">feedback</a> on this interactivity</h6>
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<mdoxml version="1.0">It is not envisaged that <a href="http://nrich.maths.org/6308">this problem</a> would be used as a class problem. It is more appropriate for an enthusiastic student or small group of students looking for a challenge to work on independently.</mdoxml>
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<p style="text-align: justify;">In the following tables we provide the numerical answers to the first four questions associated with each picture. The remaining last question is randomly generated by the program, so below we give an outline of the strategy one needs to follow to tackle it.</p>
<p style="text-align: justify;">In general, it is important to take more than one measurements of each quantity, in order to improve accuracy. Questions 1 and 2 in each problem are about making a few measurements on the picture, and question 3 involves making a simple computation using these measurements. Questions 4 and 5 are not related to the image, and only require an accurate measurement of
the scale bar and then some calculations about the scale of the picture.</p>
<p>As an example, we will solve the fifth set of questions, which are about the Pollen. Initially we measure the scale bar to find it has a length of 2.1 units.</p>
<p><strong><em>Question 1</em></strong></p>
<p><mdo:image alt="" src="Pollen%201.jpg" style="width: 547px; height: 328px;"></mdo:image></p>
<p>We find that the diameter of the particle A is 3.2 units, so we take that the radius we require is 1.6 units. To convert this into meters, we compare it with the scale bar, and so we see that the radius is</p>
<p>$$ r_A = \frac{1.6}{2.1} \times 0.00006 \approx 4.57 \times 10^{-5} $$</p>
<p>This gives us a percent error of 1.7 which is an excellent estimate. Now, with some trial and error, we find that the most accurate value we can get is $ 4.5 \times 10^{-5}$.</p>
<p> </p>
<p><strong><em>Question 2</em></strong></p>
<p><mdo:image alt="" src="Pollen%202.jpg" style="width: 545px; height: 325px;"></mdo:image></p>
<p>The diameter of particle B is 0.7 units, and since this is exactly 1 third of the scale bar, we immediately conclude that particle B has a diameter of 0.00002 and hence a radius of 0.00001 meters. Trying this value, we get a percent error of 0, so our approximation is exact.</p>
<p> </p>
<p><strong><em>Question 3</em></strong></p>
<p>First of all, we compare the volume of A to the volume of B. In particular, we get</p>
<p>$$ \frac{V_A}{V_B} = {r_A^3}{r_B^3} \approx 91.13 $$</p>
<p>So, if we assume a packing efficiency of 74%, we expect that approximately $91.13 \times 0.74 \approx 67$ B particles will fit inside a particle A. Our approximation is not absolutely correct, but again a few trials should yield the correct answer, which is 68 particles.</p>
<p> </p>
<p> </p>
<p>Now, we present the tables with the exact numerical answers to all the questions. At the end we have the strategy for solving the fifth question in each set.</p>
<p> </p>
<p><strong>Table 1: Cholera</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>0.2</td>
</tr>
<tr>
<td>2</td>
<td>0.1</td>
</tr>
<tr>
<td>3</td>
<td>24</td>
</tr>
<tr>
<td>4</td>
<td>17000</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 2: Polio</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>0.000000009</td>
</tr>
<tr>
<td>2</td>
<td>110 $\times$ 10$^6$</td>
</tr>
<tr>
<td>3</td>
<td>1 $\times$ 10$^{17}$</td>
</tr>
<tr>
<td>4</td>
<td>840000</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 3: Tobacco mosaic virus</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>0.00000029</td>
</tr>
<tr>
<td>2</td>
<td>34000</td>
</tr>
<tr>
<td>3</td>
<td>10900000000000000</td>
</tr>
<tr>
<td>4</td>
<td>100000</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 4: Mammalian mitochondria</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>0.00000055</td>
</tr>
<tr>
<td>2</td>
<td>1800</td>
</tr>
<tr>
<td>3</td>
<td>0.25</td>
</tr>
<tr>
<td>4</td>
<td>71000</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 5: Pollen</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>
<p>4.5 $\times$ 10$^{-5}$</p>
</td>
</tr>
<tr>
<td>2</td>
<td>1 $\times$ 10$^{-5}$</td>
</tr>
<tr>
<td>3</td>
<td>68</td>
</tr>
<tr>
<td>4</td>
<td>350</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 6: Escherichia Coli</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>1.6 $\times$ 10$^{-6}$</td>
</tr>
<tr>
<td>2</td>
<td>3 $\times$ 10$^{-19}$</td>
</tr>
<tr>
<td>3</td>
<td>25</td>
</tr>
<tr>
<td>4</td>
<td>16000</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 7: Human chromosomes</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>2 $\times$ 10$^{-6}$</td>
</tr>
<tr>
<td>2</td>
<td>9 $\times$ 10$^{-6}$</td>
</tr>
<tr>
<td>3</td>
<td>4.5 $\times$ 10$^7$</td>
</tr>
<tr>
<td>4</td>
<td>1600</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 8: Diatom</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>2.1 $\times$ 10$^{-5}$</td>
</tr>
<tr>
<td>2</td>
<td>9 $\times$ 10$^{9}$</td>
</tr>
<tr>
<td>3</td>
<td>3000</td>
</tr>
<tr>
<td>4</td>
<td>800</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 9: Asterionella formosa</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>1.6 $\times$ 10$^{-3}$</td>
</tr>
<tr>
<td>2</td>
<td>0.09</td>
</tr>
<tr>
<td>3</td>
<td>220</td>
</tr>
<tr>
<td>4</td>
<td>20</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 10: Drosophila melanogaster</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>2.5</td>
</tr>
<tr>
<td>2</td>
<td>0.06</td>
</tr>
<tr>
<td>3</td>
<td>1.1 $\times$ 10$^{-3}$</td>
</tr>
<tr>
<td>4</td>
<td>40</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 11: Plagiomnium affine chloroplasts</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>5.7 $\times$ 10$^{-5}$</td>
</tr>
<tr>
<td>2</td>
<td>1.5 $\times$ 10$^{-9}$</td>
</tr>
<tr>
<td>3</td>
<td>410000</td>
</tr>
<tr>
<td>4</td>
<td>700</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Table 12: Whale</strong></p>
<table border="1" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td>1</td>
<td>5.7</td>
</tr>
<tr>
<td>2</td>
<td>24</td>
</tr>
<tr>
<td>3</td>
<td>67</td>
</tr>
<tr>
<td>4</td>
<td>5 $\times$ 10$^{-3}$</td>
</tr>
</tbody>
</table>
<p> </p>
<p><strong>Outline of solution for Question 5:</strong></p>
<p>The ruler measures cm. So, the scale bar represents a length of</p>
<p>$$ L = \frac{\textrm{length in cm}\times 10^{-2}}{\textrm{Magnification}} \textrm{meters} $$</p>
<p> </p>
<p>These are the images. They are here for convenience so that they can be edited<br></br>
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<br></br>
<a href="/content/id/6308/cholera.jpg">Cholera</a><br></br>
<a href="/content/id/6308/whale.jpg">Whale</a><br></br>
<a href="/content/id/6308/polio.jpg">Polio</a><br></br>
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Scientific Measurement
Practice your skills of measurement and estimation using this interactive measurement tool based around fascinating images from biology.
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