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href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1061/functions.html">Functions</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">2.5 Exponential Models</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"> <h2><img alt="" src="../../files/MM/EM/EM-small-1.jpeg" style="width: 100px; height: 100px; float: left;">y = ka<sup>x</sup>+c</h2> <p>The exponential function is set apart from other families of functions because of the fundamental impact of having the variable in the exponent. It stops being a multiplier and starts being more 'powerful' (forgive the bad joke). These functions can be counterintuitive as the speed with which their value increases can become extraordinary. One professor famously said that 'The biggest single failing of the human race is our inability to understand the exponential function. There is also a strong link here with Geometric sequences.</p> <hr> <div class="panel panel-has-colored-body panel-turquoise"> <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-99779f92-7fff-58af-d5d6-e4a65f746a37"></span></p> <p dir="ltr"><span id="docs-internal-guid-99779f92-7fff-58af-d5d6-e4a65f746a37">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-99779f92-7fff-58af-d5d6-e4a65f746a37">recognise exponential functions</span></p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the y-intercepts of exponential functions</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">find the asymptotes of exponential functions</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"> <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>Slides Gallery</p> </div> </div> <div class="panel-body"> <div> <p>Use these slides to review the material and key points covered in the videos.</p> <div id="carousel-245" class="dynamic-gallery carousel slide" data-id="245"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-245/exponential-no-logos001.jpeg" data-fancybox="gallery-245" title="" data-caption=""><img alt="" src="../../../std-galleries/15-245/exponential-no-logos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-245/exponential-no-logos002.jpeg" data-fancybox="gallery-245" title="" data-caption=""><img alt="" src="../../../std-galleries/15-245/exponential-no-logos002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-245/exponential-no-logos003.jpeg" data-fancybox="gallery-245" title="" data-caption=""><img alt="" src="../../../std-galleries/15-245/exponential-no-logos003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-245/exponential-no-logos004.jpeg" data-fancybox="gallery-245" title="" data-caption=""><img alt="" src="../../../std-galleries/15-245/exponential-no-logos004.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-245/exponential-no-logos005.jpeg" data-fancybox="gallery-245" title="" data-caption=""><img alt="" src="../../../std-galleries/15-245/exponential-no-logos005.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-245" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-245" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-245/exponential-no-logos001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-245/exponential-no-logos002-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-245/exponential-no-logos003-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-245/exponential-no-logos004-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-245/exponential-no-logos005-thumb128.jpg"><li></li></ol> </div> </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>What are exponential functions?</p> </div> </div> <div class="panel-body"> <div> <p>This is an introduction to exponential function and all of its unique properties.<iframe allowfullscreen="" frameborder="0" height="375" mozallowfullscreen="" src="https://player.vimeo.com/video/145510312" webkitallowfullscreen="" width="500"></iframe></p> </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>Exponential y - intercepts</p> </div> </div> <div class="panel-body"> <div> <p>These are not as intuitive as they are with other functions because exponents behave differently when they are equal to zero! Watch to get a grip on this idea.</p> <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/145510980"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Understanding and finding asymptotes</p> </div> </div> <div class="panel-body"> <div> <p>Asymptotes are one of the more interesting features of exponential functions. Watch this video to understand them and how to find them.</p> <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/145511313"></iframe></div> </div> </div> <div class="panel-footer"> <div> </div> </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> <p>This section of the page can be used for quick review. The flashcards help you go over key points and the quiz lets you practice answering questions on this subtopic.</p> <h4>Flash Card</h4> <p>Review this condensed 'key point' flashcard to help you check and keep ideas fresh in your mind.</p> <div id="carousel-246" class="dynamic-gallery carousel slide" data-id="246"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-246/exponential-no-logos001-1.jpeg" data-fancybox="gallery-246" title="" data-caption=""><img alt="" src="../../../std-galleries/15-246/exponential-no-logos001-1.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-246" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-246" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-246/exponential-no-logos001-1-thumb128.jpg"><li></li></ol> </div> </div> <div class="panel-footer"> <div> </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> <h4>Self Checking Quiz</h4> <p>Practice your understanding on these quiz questions. Check your answers when you are done and read the hints where you got stuck. If you find there are still some gaps in your understanding then go back to the videos and slides above.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-318-2257"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Consider the exponetial function <img align="middle" alt="f left parenthesis x right parenthesis space equals space 2 cross times 3 to the power of x plus 1" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mo»§#215;«/mo»«msup»«mn»3«/mn»«mi»x«/mi»«/msup»«mo»+«/mo»«mn»1«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-1.php?formula=e090344d6e297a2d9342ed90edda7ecd.png">,</p><p>Fill in the corresponding values of the function for the given values of x.</p></div><div class="q-answer"><p>x = 0, so f(x) = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>x = 5, so f(x) = <input type="text" style="height: auto;" data-c="487"> <span class="review"></span> </p></div><div class="q-explanation"><p>Either substitue the value of x in to the function and calculate directly OR type the function in to your GDC and read your answers from the table. Note a<sup>0</sup>=1.</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>Consider the exponetial function <img align="middle" alt="f left parenthesis x right parenthesis space equals space minus 1.5 cross times 2 to the power of x minus 2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»1«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#215;«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«mo»-«/mo»«mn»2«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-2.php?formula=8e0171ae1701d6cce7191db314c7b5c3.png">,</p><p>Fill in the corresponding values of the function for the given values of x. (give exact answers)</p></div><div class="q-answer"><p>x = 1 so f(x) = <input type="text" style="height: auto;" data-c="-5"> <span class="review"></span> </p><p>x = -2, so f(x) = <input type="text" style="height: auto;" data-c="-2.375"> <span class="review"></span> </p></div><div class="q-explanation"><p>Either substitue the value of x in to the function and calculate directly OR type the function in to your GDC and read your answers from the table. Note a<sup>0</sup>=1.</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>Write down the y-coordinate of the y-intercpt and the equation of the asymptote for the following exponential function.</p><p><img align="middle" alt="y equals 2 cross times 1.5 to the power of x plus 3.2" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»5«/mn»«mi»x«/mi»«/msup»«mo»+«/mo»«mn»3«/mn»«mo».«/mo»«mn»2«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-3.php?formula=cb44752b08b1fd503b75d26e98c61d09.png"></p></div><div class="q-answer"><p>The y-intercept is at (0, <input type="text" style="height: auto;" data-c="5.2"> <span class="review"></span> )</p><p>The asymptote has the equation y = <input type="text" style="height: auto;" data-c="3.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>For the y - intercept, substitute x = 0. So, 1.5<sup>0 </sup>= 1 therefore y = 2 + 3.2. SInce <img align="middle" alt="2 cross times 1.5 to the power of x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»5«/mn»«mi»x«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-4.php?formula=473c9e0ceb1889f8e7f2a150ad76216e.png"> can never be zero, y can never be 3.2</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>Write down the y-coordinate of the y-intercpt and the equation of the asymptote for the following exponential function.</p><p><img align="middle" alt="y equals 1.2 cross times 3 to the power of negative x end exponent minus 18" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»1«/mn»«mo».«/mo»«mn»2«/mn»«mo»§#215;«/mo»«msup»«mn»3«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»-«/mo»«mn»18«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-5.php?formula=3185aa72b48bb6d09bb8e89c27aca522.png"></p></div><div class="q-answer"><p>The y-intercept is at (0, <input type="text" style="height: auto;" data-c="-16.8"> <span class="review"></span> )</p><p>The asymptote has the equation y = <input type="text" style="height: auto;" data-c="-18"> <span class="review"></span> </p></div><div class="q-explanation"><p>For the y - intercept, substitute x = 0. So, 3<sup>0 </sup>= 1 therefore y = 1.2 -18. SInce <img align="middle" alt="1.2 cross times 3 to the power of negative x end exponent" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mo».«/mo»«mn»2«/mn»«mo»§#215;«/mo»«msup»«mn»3«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-6.php?formula=5f0a44c4b3d8538cb3b8bfb5c258b43a.png"> can never be zero, y can never be -18</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>Consider the graphs on the axes below. Write the corresponding letter next to each function. (use upper case)</p><p style="text-align: center;"><img alt="" src="../../files/MM/EM/q1.jpg" style="width: 500px; height: 359px;"></p></div><div class="q-answer"><p>y = 2<sup>x </sup>, <input type="text" style="height: auto;" data-c="D"> <span class="review"></span> </p><p>y = 2<sup>x</sup>+1, <input type="text" style="height: auto;" data-c="A"> <span class="review"></span> </p><p>y = -2<sup>x</sup>, <input type="text" style="height: auto;" data-c="B"> <span class="review"></span> </p><p>y = 2<sup>-x</sup>, <input type="text" style="height: auto;" data-c="C"> <span class="review"></span> </p></div><div class="q-explanation"><p>You can simply put these in your calculators. BUT you should recognises that A is the same as D but moved 1 up the y - axis. C has -x in the exponent to produce that reflection and B is -2<sup>x</sup> , the negative of 2<sup>x</sup> .</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>An experiment is done to monitor how a population of bacteria grows over time. At the start, there are 200 bacteria and they triple every hour. If the function we would use to model this is in the form <img align="middle" alt="f left parenthesis x right parenthesis equals k cross times a to the power of x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»k«/mi»«mo»§#215;«/mo»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-7.php?formula=acbeb6269524b8098787e5a441d244e1.png"> , what are the values of k and a?</p></div><div class="q-answer"><p>k = <input type="text" style="height: auto;" data-c="200"> <span class="review"></span> </p><p>a = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p>200 must be the starting value, when x = 0. Next, 3 is the multiplier because the population triples for every x-increase by one.</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>The value of a car depreciates by 20% each year after purchase. It was purchased for $4000. If the function we would use to model this is in the form <img align="middle" alt="f left parenthesis x right parenthesis equals k cross times a to the power of x" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»k«/mi»«mo»§#215;«/mo»«msup»«mi»a«/mi»«mi»x«/mi»«/msup»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-7.php?formula=acbeb6269524b8098787e5a441d244e1.png"> , what are the values of k and a?</p></div><div class="q-answer"><p>k = <input type="text" style="height: auto;" data-c="4000"> <span class="review"></span> </p><p>a = <input type="text" style="height: auto;" data-c="0.8"> <span class="review"></span> </p></div><div class="q-explanation"><p>4000 must be the starting value, when x = 0. Next, 0.8 is the multiplier because the value becomes 80% of its previous value with each passing year.</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>Consider the model <img align="middle" alt="T equals 59 cross times 1.8 to the power of negative x end exponent plus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mn»59«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»8«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»+«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-8.php?formula=84419022efb953fd85a8bfe7ffdeb02c.png">, which measures temperature (T) as time (x, in minutes) passes. </p><p>What is the initial temperature at the start of the experiment? </p></div><div class="q-answer"><p>T = <input type="text" style="height: auto;" data-c="74"> <span class="review"></span> </p></div><div class="q-explanation"><p>This is when x = 0.</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>Consider the model <img align="middle" alt="T equals 59 cross times 1.8 to the power of negative x end exponent plus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mn»59«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»8«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»+«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-8.php?formula=84419022efb953fd85a8bfe7ffdeb02c.png">, which measures temperature (T) as time (x, in minutes) passes. </p><p>What is the temperature after 4 minutes? (3sf)</p></div><div class="q-answer"><p>T = <input type="text" style="height: auto;" data-c="20.6"> <span class="review"></span> </p></div><div class="q-explanation"><p>This is when x = 4. Read this from the table on your GDC.</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>Consider the model <img align="middle" alt="T equals 59 cross times 1.8 to the power of negative x end exponent plus 15" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»T«/mi»«mo»=«/mo»«mn»59«/mn»«mo»§#215;«/mo»«mn»1«/mn»«mo».«/mo»«msup»«mn»8«/mn»«mrow»«mo»-«/mo»«mi»x«/mi»«/mrow»«/msup»«mo»+«/mo»«mn»15«/mn»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-8.php?formula=84419022efb953fd85a8bfe7ffdeb02c.png">, which measures temperature (T) as time (x, in minutes) passes. </p><p>After how many minutes does the temperature fall below 18 degrees? (nearest minute)</p></div><div class="q-answer"><p>After <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> minutes.</p></div><div class="q-explanation"><p>Read this from the table on your GDC.</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> </section> </div> </div> <div class="panel-footer"> <div> </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> <p>Exam Style Questions</p> </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><img alt="" src="../../files/MM/EM/EMQ1.jpg" style="width: 700px; height: 282px;"></p> <p>Video Solution</p> </div> </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><img alt="" src="../../files/MM/EM/EMQ2(1).jpg" style="width: 700px; height: 284px;"></p> <p>Video Solution</p> <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/592617474"></iframe></div> </section> </div> </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 3</p> </div> </div> <div class="panel-body"> <p><img alt="" src="../../files/MM/EM/EMQ3.jpg" style="width: 700px; height: 356px;"></p> <p>Video Solution</p> <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/592617873"></iframe></div> </section> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <p dir="ltr"> </p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p dir="ltr"><span id="docs-internal-guid-31d13d15-7fff-d54c-9238-6daefe6237e1">Isabella carries out an experiment in her biology lesson about the growth of a bacteria. She thinks that the growth can be modelled with an exponential function</span></p> <p dir="ltr" style="text-align: center;"><span class="math-tex">\(P\left(t\right)=Ae^{kt}\)</span></p> <p dir="ltr"><span id="docs-internal-guid-31d13d15-7fff-d54c-9238-6daefe6237e1">Where P is the area covered by the mould at time, t in hours since the start of the experiment and A and k are constants. </span></p> <p dir="ltr"><span id="docs-internal-guid-31d13d15-7fff-d54c-9238-6daefe6237e1">The area covered by the mould at the start of the experiment is 400, and after 6 hours the area covered is 2400. </span><span id="docs-internal-guid-31d13d15-7fff-d54c-9238-6daefe6237e1"></span></p> <p dir="ltr">a. Find the value of A </p> <p dir="ltr">b. Find the value of k</p> <p dir="ltr"><span id="docs-internal-guid-31d13d15-7fff-d54c-9238-6daefe6237e1">c. Find the area covered by mould after 10 hours </span></p> <p dir="ltr">Video Solution</p> </section> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> <p> </p> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <p style="text-align: center;"><br> </p> <h4> </h4> <p>.....<span id="cke_bm_805E" style="display: none;"> </span></p> <div class="page-container panel-self-assessment" data-id="2257"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>2.5 Exponential Models</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" 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