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class="page_title"> 1.6 Accuracy and Estimation <a href="#" class="mark-page-favorite pull-right" data-pid="2121" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1057/number-and-algebra.html">Number and Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">1.6 Accuracy and Estimation</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <div class="intro-card"> <div class="bg-cover" style="background-image: url('/media/mathsapplications/football-crowd.png');"> </div> <img src="../../football-crowd-1.png" style="display: none"> <div class="content"> <p class="theading">Accuracy and Estimation </p> <p class="text">When numbers are reported in the news, or as statistical results, they are either displayed exactly, or they are rounded to a certain degree of accuracy. Understanding how and when to do this can give us a deeper insight into the important numbers we run into in our daily lives. When numbers are estimated, it leads to a lack of accuracy, the size of which can be represented by the percentage error. We will also learn how to find this in this chapter. </p> </div> </div> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key concepts</p> </div> </div> <div class="panel-body"> <p><span id="docs-internal-guid-31acc368-7fff-9377-8a56-c5cb9b8fc118"></span></p> <p dir="ltr"><span id="docs-internal-guid-31acc368-7fff-9377-8a56-c5cb9b8fc118">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation"><span id="docs-internal-guid-31acc368-7fff-9377-8a56-c5cb9b8fc118">Round using decimal places and significant figures</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Find the upper and lower bounds of rounded numbers</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Estimate the result of a calculation using rounding</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">Find the percentage error of a calculation</p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following is a series of videos that will help you understand, learn about and review this sub-topic..</p> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Rounding using decimal places</p> </div> </div> <div class="panel-body"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/455197229"></iframe></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Rounding using significant figures</p> </div> </div> <div class="panel-body"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/455202494"></iframe></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Upper and Lower Bounds</p> </div> </div> <div class="panel-body"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/666470159?h=44e0db9a0f&badge=0&autopause=0&player_id=0&app_id=58479"></iframe></div> <script src="../../../api/player.js"></script></div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>4. Estimation</p> </div> </div> <div class="panel-body"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/457221963"></iframe></div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>5. Percentage Error</p> </div> </div> <div class="panel-body"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/457221515"></iframe></div> </div> <div class="panel-footer"> <div> <p><a class="expander" href="#"><span class="fa fa-plus"></span></a></p> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <ul class="gallery layout-row thumb48"><li class=""><a href="../../../std-galleries/15-227/161.png" data-fancybox="gallery-227" class="fancy"><img src="../../../std-galleries/15-227/161-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-227/162.png" data-fancybox="gallery-227" class="fancy"><img src="../../../std-galleries/15-227/162-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-227/163.png" data-fancybox="gallery-227" class="fancy"><img src="../../../std-galleries/15-227/163-thumb128.jpg" class="thumb"></a></li><li class=""><a href="../../../std-galleries/15-227/164.png" data-fancybox="gallery-227" class="fancy"><img src="../../../std-galleries/15-227/164-thumb128.jpg" class="thumb"></a></li><li><a href="../../../std-galleries/15-227/165.png" data-fancybox="gallery-227" class="fancy"><img src="../../../std-galleries/15-227/165-thumb128.jpg" class="thumb"></a></li></ul> </section> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <div> <p>Self Checking Quiz</p> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the worked solutions when you get stuck. If you find there are still some gaps in your understanding then go back to the videos above<em>.</em></p> <div class="tib-quiz" data-stats="15-515-2121"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 14.7854 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>15</span></label> </p><p><label class="radio"> <input type="radio"> <span>14.785</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>14.8</span></label> </p><p><label class="radio"> <input type="radio"> <span>14.7</span></label> </p></div><div class="q-explanation"><p>14.8 - the third non zero digit is 7, which is rounded up to 8 because of the 8 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 0.003046 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"><input type="radio">0.003</label></p><p><label class="radio"><input type="radio"><span>0</span></label> </p><p><label class="radio"> <input type="radio">0.00304</label></p><p><label class="radio"> <input class="c" type="radio"> 0.00305</label></p></div><div class="q-explanation"><p>0.00305 - the third non zero digit is 4, which is rounded up to 7 because of the 6 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 12348 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"><input type="radio">12350</label></p><p><label class="radio"> <input type="radio">12400</label></p><p><label class="radio"><input type="radio">123</label></p><p><label class="radio"> <input class="c" type="radio"> 12300</label></p></div><div class="q-explanation"><p>12300 - the third non zero digit is 3, which stays the same because of the 4 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 6504905 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"> <input type="radio">6510000</label></p><p><label class="radio"><input type="radio">650 </label></p><p><label class="radio"><input type="radio">6505000</label></p><p><label class="radio"> <input class="c" type="radio"> 6500000</label></p></div><div class="q-explanation"><p>6500000 - the third non zero digit is 0, which stays the same because of the 4 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is 452756 correctly rounded to 3 significant figures</p></div><div class="q-answer"><p><label class="radio"><input type="radio">453 </label></p><p><label class="radio"><input type="radio">450000</label></p><p><label class="radio"> <input class="c" type="radio"> 453000</label></p><p><label class="radio"> <input type="radio">452000</label></p></div><div class="q-explanation"><p>453000 - the third non zero digit is 2, which becomes 3 because of the 7 that follows</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>A rectangular field measures 23m by 63m, correct to the nearest metre. What is the smallest possible area of the field in square metres? (Give the exact answer)</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>1414.76</span></label> </p><p><label class="radio"> <input type="radio"> <span>1417.50</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>1406.25</span></label> </p><p><label class="radio"> <input type="radio"> <span>1449</span></label> </p></div><div class="q-explanation"><p>The lower limt of both measurements are 22.5 and 62.5.</p><p>22.5 x 62.5 = 1406.25</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>A rectangular field measures 15m by 21m, correct to the nearest metre. What is the upper limit for the area of the field in square metres? (Round your answer to 3sf))</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>297</span></label></p><p><label class="radio"> <input class="c" type="radio"> <span>333</span></label></p><p><label class="radio"> <input type="radio"> <span>332</span></label></p><p><label class="radio"> <input type="radio"> <span>315</span></label></p></div><div class="q-explanation"><p>The upper limt of both measurements are 15.5 and 21.5.</p><p>15.5 x 21.5 = 333.25, which is 333 when rounded to 3sf</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A bucket is used to lower the level of water in a swimming pool - 100 full buckets were removed. The bucket is advertised with a capcity of 3 litres, The actual capacity of the bucket is 3.2 litres. What is the percentage error if the marketed capacity is used to calculate the volume of water that is removed? </p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>6.25%</span></label> </p><p><label class="radio"> <input type="radio"> <span>6.30%</span></label> </p><p><label class="radio"> <input type="radio"> <span>6.67%</span></label> </p><p><label class="radio"> <input type="radio"> <span>6.5%</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(Volume\quad using\quad marketing\quad capacity\quad =\quad 300l\\ Actual\quad volume\quad =\quad 320l\\ Percentage\quad error\quad =\quad \frac { 320-300 }{ 320 } \times 100=6.25\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Hannah estimates that the sum of the first 20 terms of the following sequence, <span class="math-tex">\({ U }_{ n }=12+8(n-1)\)</span>, is 2000. What is her percentage error</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>12%</span></label> </p><p><label class="radio"> <input type="radio"> <span>13.64%</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>13.6%</span></label> </p><p><label class="radio"> <input type="radio"> <span>12.0%</span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\({ U }_{ n }=12+8(n-1)\\ n=20,\quad { U }_{ 1 }=12,\quad d=8\\ S_{ 20 }=\frac { 20 }{ 2 } (24+8(20-1))=1760\\ Error\quad =\quad \frac { 2000-1760 }{ 1760 } \times 100=13.6\quad (3sf)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Nick converted 300 Zambia Kwechas (ZMW) to Euros (EUR) and the bank were offering an exchange rate of 1 ZMW = 0.049 EUR. If he converted 300 and got 14 EUR, how much commission was he charged?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>6.7%</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>4.76%</span></label> </p><p><label class="radio"> <input type="radio"> <span>4.75%</span></label> </p><p><label class="radio"> <input type="radio"> <span>2.4%</span></label> </p></div><div class="q-explanation"><p>The conversion rate is</p><p><span class="math-tex">\(1\quad ZMW\quad =\quad 0.049\quad EUR\)</span></p><p>so converting 300 ZMW gives:</p><p><span class="math-tex">\(300\quad ZMW\quad =\quad 14.7\quad EUR\)</span></p><p>Buzz got 14 so he paid 0.7 EUR in commission, making the percentage difference:</p><p><span class="math-tex">\(\frac{0.7}{14.7}\cdot100=4.76\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <h4>Exam Style Questions</h4> </div> </div> <div class="panel-body"> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <p dir="ltr"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08"></span></p> <p dir="ltr"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08">Given <span class="math-tex">\(a=b-\frac{\sqrt{c}}{d}\)</span></span>, b = 1.775, c = 1.347 and d = 32,</p> <p dir="ltr"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08">(a) calculate the value of a.</span></p> <p dir="ltr"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08">Luisa first writes b, c and d correct to one significant figure and then uses these values to estimate the value of a.</span></p> <p dir="ltr"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08">(b) (i) Write down b, c and d each correct to one significant figure.</span></p> <p dir="ltr"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08"> (ii) Write down Luisa’s estimate of the value of a.</span></p> <p dir="ltr"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08">(c) Calculate the percentage error in Luisa’s estimate of the value of p.</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08">(Total 6 marks)</span></p> <div><span id="docs-internal-guid-9ac01a28-7fff-a962-6788-6f0dd9cebc08"></span></div> </div> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598286087?badge=0&autopause=0&player_id=0&app_id=58479&h=94487ef888"></iframe></div> <script src="../../../api/player.js"></script></section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <p dir="ltr"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">The volume of a sphere is <span class="math-tex">\(V=\sqrt{\frac{S^{3}}{36\pi}}\)</span></span> , where S is its surface area.</p> <p dir="ltr"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">The surface area of a sphere is <span class="math-tex">\(300cm^2\)</span></span>.</p> <p dir="ltr"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">(a) Calculate the volume of the sphere. Give your answer correct to two decimal places.</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">(3)</span></p> <p dir="ltr"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">(b) Write down your answer to (a) correct to the nearest integer.</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">(1)</span></p> <p dir="ltr"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">(c) Calculate the percentage error between the exact volume and your answer to part b</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">(3)</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-959925d0-7fff-9c6f-ff83-ab3efe676e56">(Total 6 marks)</span></p> </div> </div> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598286087?badge=0&autopause=0&player_id=0&app_id=58479&h=94487ef888"></iframe></div> <script src="../../../api/player.js"></script></section> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <p dir="ltr"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">Given that <span class="math-tex">\(x=\frac{\left(\sqrt[3]{p}-d\right)}{52}\)</span></span> ,</p> <p dir="ltr"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(a) Calculate the exact value of x when p = 0.4136 and d = 0.00025.</span></p> <p dir="ltr" style="margin-left: 36pt; text-align: right;"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(2)</span></p> <p dir="ltr"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(b) Write down the answer to part (a) correct to three decimal places.</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(1)</span></p> <p dir="ltr"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(c) Write down the answer to part (a) correct to three significant figures.</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(1)</span></p> <p dir="ltr"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(d) Write down the answer to part (a) in the form a × 10</span><sup>k</sup>, where 1 ≤ a < 10</p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(2)</span></p> <p dir="ltr" style="text-align: right;"><span id="docs-internal-guid-aa66d5f9-7fff-8b50-bae7-fcc228591b58">(Total 6 marks)</span></p> </div> </div> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598295385?badge=0&autopause=0&player_id=0&app_id=58479&h=b35e00b4e3"></iframe></div> <script src="../../../api/player.js"></script></section> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2121"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label 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