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Applications of trigonometry</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../1862/34-circles.html">3.4 Circles</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2120/35-36-perpendicular-bisectors-voronoi-diagrams-.html"> 3.5 & 3.6 Perpendicular Bisectors & Voronoi Diagrams </a></label></li></ul></li><li class=""><label style="padding-left: 0px"><a class="expander" href="#" style="font-size: .9em"><i class="fa fa-fw fa-caret-right"></i></a><a href="../1065/statistics-and-probability.html">Statistics and Probability</a></label><ul class="side-nav level-1"><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2188/41-statistical-concepts.html">4.1 Statistical concepts</a></label></li><li class=""><label style="padding-left: 14px"><i class="fa fa-fw"></i><a href="../2099/42-43-cumulative-frequency-box-plots-.html">4.2 & 4.3 Cumulative Frequency & Box Plots 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here</span></li> </ol> <article id="main-article"> <h2>Mathematics Applications & Interpretation SL (standard level)</h2> <h4 style="text-align: center;"><strong><em>- Your one stop virtual teacher, practice zone & revision guide! -</em></strong></h4> <p>Welcome to this study IB site for students of the <meta charset="utf-8"><b id="docs-internal-guid-dd8d1917-7fff-0f50-27f7-f05fd6c0659e">IB Mathematics Applications and Interpretation</b> <strong>SL</strong> (standard level) course. This short page is designed as a brief introduction to the site to help you understand what is here and how you can use it. The site is designed to be a 'virtual teacher' and a 'practice and revision center'. The virtual teacher aspect is a huge bank of slides and videos aimed at explaining the key concepts of the course. These are then backed up by revision flashcards, hundreds of practice questions with advice and feedback and exam style questions with full worked solutions. In short, everything you need to support the studying and assessment of this course.</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/672064558"></iframe></div> <p style="text-align: center;"><img alt="" src="../../images/2022-01-09_13-28-49.jpg" style="width: 700px; height: 524px;"></p> <div class="wine"> <h2>Have a look</h2> <p>With no sign up at all you can see and use this page on the <a href="../2191/49-normal-distribution.html" title="4.9 Normal Distribution">4.9 Normal Distribution</a> as an example of all the features our pages offer.</p> <p>If you create a basic account at no cost, you can see and use all our <a href="../1054/your-graphical-display-calculator.html" title="Your Graphical Display Calculator">Your Graphical Display Calculator</a> pages for <a href="../1055/casio-.html" title="Casio ">Casio, </a><a href="../1056/ti-84.html" title="TI-84">TI-84</a> and <a href="../2081/ti-nspire.html" title="TI-Nspire">TI-Nspire</a> 4 further chapters..</p> <p> <a href="../2117/11-standard-index-form-.html" title="1.1 Standard Index Form ">1.1 Standard Index Form </a></p> <p> <a href="../2186/25-trigonometric-models-.html" title="2.5 Trigonometric Models ">2.5 Trigonometric Models </a></p> <p> <a href="../2398/31-volume-and-surface-area.html" title="3.1 Volume and surface area">3.1 Volume and surface area</a></p> <p> <a href="../2400/51-53-introduction-to-calculus.html" title="5.1 & 5.3 Introduction to Calculus">5.1 & 5.3 Introduction to Calculus</a></p> <p>That is already a lot of useful resources to help you get a feel for our site an dpersuade you to pay the very reasonable fee for the rest!</p> </div> <hr> <div class="greenBg"> <p><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6"></span></p> <h2>What is here?</h2> <p>This site is explicitly for students of the <em><strong>Maths Applications & Interpetation SL</strong></em> (standard level) course and includes....</p> <ul dir="ltr"> <li><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6"><strong>200+ Teaching videos</strong> covering the key concepts on the syllabus.</span></li> <li><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6"><strong>200+ Slides</strong> with visual explanations and examples.</span></li> <li><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6"><strong>Hundreds of onscreen practice questions</strong> with feedback on each topic</span></li> <li>A <em><strong>questionbank with 1700+ questions with answers and explanations</strong></em> where you can choose a set to practice on by number of questions, topic, subtopic and difficulty level.</li> <li><strong><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6">Revision cards</span></strong></li> <li><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6"><strong>100+ original</strong> exam style questions with video solutions</span></li> <li><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6">Regular updates with more videos and questions going up every month.</span></li> </ul> <h3>And more.....</h3> <ul> <li>You can also find a comprehensive section on how to use your <a href="../1054/your-graphical-display-calculator.html" title="Your Graphical Display Calculator">Your Graphical Display Calculator</a></li> <li>There is a really helpful chapter on how to consider your <a href="../1083/ia-technology.html" title="IA & technology">IA & technology</a></li> <li>A crucial section on <a href="../1085/exam-advice.html" title="Exam Advice">Exam Advice</a></li> <li>With a subsection on how to get <a href="../1084/sixes-sevens.html" title="Sixes & sevens">Sixes & sevens</a></li> </ul> <div><span id="docs-internal-guid-78d6c124-7fff-525a-8956-e3b0de89a4d6"></span></div> </div> <div class="blueBg"> <h3>Video Introduction</h3> <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/666508765?h=00d3e8d919&badge=0&autopause=0&player_id=0&app_id=58479"></iframe></div> <script src="../../../api/player.js"></script></div> <div class="pinkBg"> <h3>Teaching Videos</h3> <p><span id="docs-internal-guid-6e715e81-7fff-5538-f779-c9de05931317">These allow you to review key syllabus concepts that you have learned in class. Slow, careful demonstrations and explanations that you can watch again and again. Designed to make you think and help you understand and improve. Some are designed to explain concepts from the beginning and others are focussed on very specific skills</span></p> <div class="panel panel-has-footer panel-fucsia"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Example</p> </div> </div> <div class="panel-body"> <p>This video introduces the idea of data sets and distributions leading to the key properties.</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/589841155"></iframe></div> </div> <div class="panel-footer"> </div> </div> </div> <div class="yellowBg"> <h3>Teaching Slides</h3> <p>Each topic page starts with a gallery of teaching slides that can be easily scanned so that you can find a slide that offers you a quick explanation, if you don't want to watch the whole video. Like flicking through the pages of a book, this section is desigend to be a quick reference guide to what you are looking for. You can click on any image to make it fill the screen and then swipe through them - this works really well on a mobile device.</p> <div class="panel panel-has-footer panel-yellow"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Example Teaching Slides</p> </div> </div> <div class="panel-body"> <div> <p>This gallery can tell you what you need to know about volume and surface area of pyramids and cones</p> <div id="carousel-258" class="dynamic-gallery carousel slide" data-id="258"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-258/vsa-part-3-nologos001.jpeg" data-fancybox="gallery-258" title="" data-caption=""><img alt="" src="../../../std-galleries/15-258/vsa-part-3-nologos001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-258/vsa-part-3-nologos002.jpeg" data-fancybox="gallery-258" title="" data-caption=""><img 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practice. You can get instant feedback on your answers including advice about how to tackle the question and worked solutions. See the example below.</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#6009a310"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="6009a310"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Calculus Part 3 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="15-330-1053" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Question 1</p><p>Consider the function <span class="math-tex">\(f(x)={ x }^{ 2 }-4x+7\)</span></p><p>then <span class="math-tex">\(f'(x)\quad =\quad ax\quad +\quad b\)</span></p><p>a) What are the values of a and b?</p><p>b) What is the value of x when the gradient is zero.</p><p>c) And so if the coordinates of the vertex are (x,y), what are the values of x and y?</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span> </p><p>b) x = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> </p><p>c) x = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> , y = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) Function differentiates to <span class="math-tex">\(f'(x)\quad =\quad 2x-4\)</span> (careful, since it is -4, but the equation says +b, then b must be negative.</p><p>b) <span class="math-tex">\(when\quad gradient\quad =\quad 0,\\ f'(x)\quad =\quad 2x-4\quad =\quad 0\\ so,\quad 2x\quad =\quad 4\\ and\quad x\quad =\quad 2\)</span></p><p>c) <span class="math-tex">\(At\quad vertex,\quad f'(x)\quad =\quad 0\quad so\quad x\quad =\quad 2\\ Substitute\quad x\quad =\quad 2\quad into\quad f(x)={ x }^{ 2 }-4x+7\\ So,\quad y\quad =\quad 3\)</span> (You might use your table function here)</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 2</p><p>Consider the function <span class="math-tex">\(f(x)={ x }^{ 2 }+6x+3\)</span></p><p>then <span class="math-tex">\(f'(x)\quad =\quad ax\quad +\quad b\)</span></p><p>a) What are the values of a and b?</p><p>b) What is the value of x when the gradient is zero.</p><p>c) And so if the coordinates of the vertex are (x,y), what are the values of x and y?</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> </p><p>b) x = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> </p><p>c) x = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> , y = <input type="text" style="height: auto;" data-c="-6"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) Function differentiates to <span class="math-tex">\(f'(x)\quad =\quad 2x+6\)</span> </p><p>b) <span class="math-tex">\(when\quad gradient\quad =\quad 0,\\ f'(x)\quad =\quad 2x+6\quad =\quad 0\\ so,\quad 2x\quad =\quad -6\\ and\quad x\quad =\quad -3\)</span></p><p>c) <span class="math-tex">\(At\quad vertex,\quad f'(x)\quad =\quad 0\quad so\quad x\quad =\quad -3\\ Substitute\quad x\quad =\quad -3\quad into\quad f(x)={ x }^{ 2 }+6x+3\\ So,\quad y\quad =\quad -6\)</span> (You might use your table function here)</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 3</p><p>Consider the function <span class="math-tex">\(f(x)=\frac { 1 }{ 3 } { x }^{ 3 }-\frac { 1 }{ 2 } { x }^{ 2 }-12x+4\)</span></p><p>Then <span class="math-tex">\(f'(x)=a{ x }^{ 2 }+b{ x }+c\)</span></p><p>a) What are the values of a, b and c?</p><p>b) What are the values of x when the gradient is zero? (Give your answers in numerical order)</p><p>c) What are the coordinates of the local maximum and minimum for this function</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="-12"> <span class="review"></span> </p><p>b) x = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> , x = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p><p>c) Local maximum x = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span> , y = <input type="text" style="height: auto;" data-c="26.5"> <span class="review"></span> (give y to 3sf)</p><p>Local Minimum, x = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> , y = <input type="text" style="height: auto;" data-c="-30.7"> <span class="review"></span> (give y to 3sf)</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(f'(x)={ x }^{ 2 }-{ x }-12\)</span>(Pay attention to the negative signs)</p><p>b) <span class="math-tex">\(When\quad the\quad gradient\quad is\quad zero\\ f'(x)={ x }^{ 2 }-{ x }-12\quad =\quad 0\\ f'(x)=(x+3)(x-4)\quad =\quad 0\\ x\quad =\quad -3\quad or\quad 4\)</span> You can use the polynomial solver on your GDC to solve this equation</p><p>c) Read the gradient function from your GDC for x = -3 and x = 4</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 4</p><p>Consider the function <span class="math-tex">\(f(x)={ 2x }^{ 2 }+\frac { 32 }{ x } \)</span></p><p>a) This can be written as, <span class="math-tex">\(f(x)={ 2x }^{ 2 }+m{ x }^{ n }\)</span>, what are the values of m and n</p><p>b) The derivative of the function is given by <span class="math-tex">\(f'(x)\quad =\quad ax+b{ x }^{ c }\)</span>what are the values of a, b and c?</p><p>c) At what value of x is the gradient = 0?</p><p>d) What are the coordiantes of this local minimum?</p></div><div class="q-answer"><p>a) m = <input type="text" style="height: auto;" data-c="32"> <span class="review"></span> , n = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> </p><p>b) a = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="-32"> <span class="review"></span> , c = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> </p><p>c) x = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> </p><p>d) x = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> , y = <input type="text" style="height: auto;" data-c="24"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) BEWARE you are NOT differentiating here, you are just expressing it as a negative index. Remember...</p><p><span class="math-tex">\(\frac { 1 }{ { x }^{ n } } ={ x }^{ -n }\quad and\quad \frac { 3 }{ { x }^{ n } } ={ 3x }^{ -n }\quad and\quad \frac { 2 }{ { 5x }^{ n } } =\frac { 2 }{ 5 } { x }^{ -n }\)</span> so <span class="math-tex">\(f(x)={ 2x }^{ 2 }+\frac { 32 }{ x } \quad =\quad f(x)={ 2x }^{ 2 }+32{ x }^{ -1 }\)</span></p><p>...</p><p>b) <span class="math-tex">\(if,\quad f(x)={ 2x }^{ 2 }+32{ x }^{ -1 },\quad then,\quad f'(x)\quad =\quad 4x-32{ x }^{ -2 }\)</span></p><p>...</p><p>c) <span class="math-tex">\(when\quad gradient\quad =\quad 0,\\ f'(x)\quad =\quad 4x-32{ x }^{ -2 }\quad =\quad 0\\ f'(x)\quad =\quad 4x-\frac { 32 }{ { x }^{ 2 } } =0\\ so\quad 4x\quad =\quad \frac { 32 }{ { x }^{ 2 } } \\ and,\quad 4{ x }^{ 3 }\quad =\quad 32\\ and\quad { x }^{ 3 }\quad =\quad 8,\quad \\ so\quad x\quad =\quad 2\)</span> Again, your equation solver on the GDC can do this for you.</p><p>...</p><p>d) <span class="math-tex">\(At\quad vertex,\quad f'(x)\quad =\quad 0\quad so\quad x\quad =\quad 2\\ Substitute\quad x\quad =\quad 2\quad into\quad f(x)={ 2x }^{ 2 }+\frac { 32 }{ x } \\ So,\quad y\quad =\quad 24\)</span> Again, the table function will help you to avoid mistakes</p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 5</p><p>The Volume of water (<em>V</em>) in a container varies with some given conditions against time (<em>t</em>), The Volume, <span class="math-tex">\(V{ cm }^{ 3 }\)</span> , is given by <span class="math-tex">\(V=400+3t-{ t }^{ 2 }\)</span></p><p>a) If <span class="math-tex">\(\frac { dV }{ dt } =at+b\)</span> then what are the values of a and b?</p><p>b) At what value of t will the volume be at a maximum?</p><p>c) What will the maximum volume be?</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>b) t = <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span> (give the exact answer as a decimal)</p><p>c) Maximum Volume = <input type="text" style="height: auto;" data-c="402"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <span class="math-tex">\(if\quad V=400+3t-{ t }^{ 2 }\quad then,\frac { dV }{ dt } =3-2t\)</span></p><p>b) <span class="math-tex">\(at\quad maximum,\quad \frac { dV }{ dt } =3-2t=0\\ so,\quad 3=2t\\ and\quad t=\frac { 3 }{ 2 } =1.5\)</span></p><p>c) <span class="math-tex">\(Maximum\quad occurs\quad at\quad t\quad =\quad 1.5\\ when\quad t=1.5\\ V=400+3(1.5)-{ (1.5) }^{ 2 }\\ V=400+4.5-2.25\\ V=402.25,\quad V=402{ cm }^{ 3 }\quad (3sf)\)</span> </p><p>Using the GDC, if you enter the Volume function and have the derivative switched on then you can see that at t = 1.5, Volume is 402.25 and the derivative is zero.</p><p><img alt="" src="../../files/CLS/spo/q5.jpg" style="width: 302px; height: 141px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 6</p><p>A company's profit, P, in 1000s of euros can be modelled by the the function, <span class="math-tex">\(P(x)=19x-0.07{ x }^{ 2 }-10\)</span>where <span class="math-tex">\(x\)</span>, is a measure of units of product for the company.</p><p>a) if <span class="math-tex">\(\frac { dP}{ dx } =ax+b,\)</span>what are the values of a and b?</p><p>b) At what value of x while the profit be at a maximum?</p><p>c) Use the rounded answer to part b) to calculate the value of the maximum profit.</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="-0.14"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="19"> <span class="review"></span> </p><p>b) x = <input type="text" style="height: auto;" data-c="136"> <span class="review"></span> (give your answer to 3sf)</p><p>c) Maximum profit, P = <input type="text" style="height: auto;" data-c="1280"> <span class="review"></span> (give your answer as the number of 1000s to 3sf)</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(if\quad P(x)=19x-0.07{ x }^{ 2 }-10,\quad then,\frac { dP }{ dx } =19-0.14x\)</span></p><p>b) <span class="math-tex">\(At\quad maximum,\quad \frac { dP }{ dx } =19-0.14x=0\\ 19=0.14x\\ x=\frac { 19 }{ 0.14 } =135.71428.....\\ x=\quad 136\quad (3sf)\)</span></p><p>c) <span class="math-tex">\(maximum\quad profit\quad occurs\quad at\quad x\quad =\quad 136\\ Substituting\quad in\quad to\quad P(x)=19x-0.07{ x }^{ 2 }-10\\ gives\quad V=1279.2\\ V=1280\quad (3sf)\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 7</p><p>Consider the formula A = bh which is subject to the constraint that b - h=10. </p><p>a) Which of the expressions is the correct expression for A in terms of b?</p><p> <label class="radio"><input type="radio"> <span class="math-tex">\(A=bh\)</span></label> <label class="radio"><input class="c" type="radio"> <span class="math-tex">\(A={ b }^{ 2 }-10b\)</span></label> <label class="radio"><input type="radio"> <span class="math-tex">\(A=10{ b }^{ 2 }-b\)</span></label> <label class="radio"><input type="radio"> <span class="math-tex">\(A=10{ b }\)</span></label></p><p>b) If <span class="math-tex">\(\frac { dA }{ db } =mb+n\)</span>, what are the values of m and n?</p><p>c) For what value of b is A a minimum?</p><p>d) What is the minimum value of A?</p></div><div class="q-answer"><p>b) m = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> , n = <input type="text" style="height: auto;" data-c="-10"> <span class="review"></span> </p><p>c) b = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span> </p><p>d) Minimum value of A = <input type="text" style="height: auto;" data-c="-25"> <span class="review"></span> </p></div><div class="q-explanation"><p>a) <span class="math-tex">\(A=bh\\ b-h=10\quad so\\ b=10+h\quad and\\ b-10=h\\ Substitute\quad h\quad =\quad b-10\\ A=b(b-10)\\ A={ b }^{ 2 }-10b\)</span></p><p>b) <span class="math-tex">\(if\quad A={ b }^{ 2 }-10b\\ then\quad \frac { dA }{ db } =2b-10\)</span></p><p>c) <span class="math-tex">\(at\quad minimum\quad value,\quad \frac { dA }{ db } =2b-10=0\\ so\quad 2b=10\\ and\quad b=5\)</span></p><p>d) <span class="math-tex">\(Substitute\quad b=5\quad into\quad A={ b }^{ 2 }-10b\\ so\quad A=-25\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 8</p><p>Consider the formula <span class="math-tex">\(A={ x }^{ 3 }+2xh\)</span>, subject to the condition that <span class="math-tex">\(h=\frac { 3 }{ { x }^{ 3 } } \)</span></p><p>a) A can be expressed in terms of x only like this, <span class="math-tex">\(A={ x }^{ 3 }+p{ x }^{ q }\)</span>, what are the values of p and q?</p><p>b) <span class="math-tex">\(\frac { dA }{ dx } =m{ x }^{ 2 }-n{ x }^{ -3 }\)</span>, what are the values of m and n?</p><p>c) For what value of x is A at a local minimum?</p></div><div class="q-answer"><p>a) p = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span> , q = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span> </p><p>b) m = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> , n = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span> </p><p>c) x = <input type="text" style="height: auto;" data-c="1.32"> <span class="review"></span> (Answer to 3sf)</p></div><div class="q-explanation"><p>a) Substitute the value for h in to the expression for A,</p><p><span class="math-tex">\(if\quad A={ x }^{ 3 }+2xh\quad and\quad h=\frac { 3 }{ { x }^{ 3 } } \\ Substituting\quad gives\\ A={ x }^{ 3 }+\frac { 2x\times 3 }{ { x }^{ 3 } } \\ A={ x }^{ 3 }+\frac { 6x }{ { x }^{ 3 } } \\ A={ x }^{ 3 }+\frac { 6 }{ { x }^{ 2 } } \\ A={ x }^{ 3 }+6{ x }^{ -2 }\)</span></p><p>b) Differentiate A with respect to x</p><p><span class="math-tex">\(if\quad A={ x }^{ 3 }+6{ x }^{ -2 }\\ \\ then,\frac { dA }{ dx } =3{ x }^{ 2 }-12{ x }^{ -3 }\)</span></p><p>c) Solve for gradient = 0</p><p><span class="math-tex">\(At\quad minimum,\quad \frac { dA }{ dx } =3{ x }^{ 2 }-12{ x }^{ -3 }=0\\ 3{ x }^{ 2 }-12{ x }^{ -3 }=0\quad (use\quad GDC\quad solver)\\ x=1.3195....\\ x=1.32\quad (3sf)\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 9</p><p>A closed cuboid box has Volume, <span class="math-tex">\(V={ x }^{ 2 }h\)</span>. Its surface area, A, is limted to <span class="math-tex">\(600{ cm }^{ 2 }\)</span></p><p><img alt="" src="../../files/CLS/spo/q8.jpg" style="width: 150px; height: 199px;">Dimensions shown are in cm</p><p>a) A is given by <span class="math-tex">\(A=m{ x }^{ 2 }+nxh\)</span>, what are the values of m and n?</p><p>b) Since A = 600, we can say that <span class="math-tex">\(\frac { a-{ x }^{ 2 } }{ bx } =h\)</span>, what are the values of a and b?</p><p>c) Use this expression for h to express the volume in terms of h only, where <span class="math-tex">\(V=px-\frac { 1 }{ 2 } { x }^{ q }\)</span>, what are the values of p and q?</p><p>d) For what value of x is the Volume at a maximum?</p><p>e) What is the maximum volume?</p><p>f) What will the value of h be at this volume?</p></div><div class="q-answer"><p>a) m = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> , n = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span> </p><p>b) a = <input type="text" style="height: auto;" data-c="300"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> </p><p>c) p = <input type="text" style="height: auto;" data-c="150"> <span class="review"></span> , q = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span> </p><p>d) x = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> cm</p><p>e) Maximum Volume = <input type="text" style="height: auto;" data-c="1000"> <span class="review"></span> <span class="math-tex">\({ cm }^{ 3 }\)</span></p><p>f) h = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> cm</p></div><div class="q-explanation"><p>a) <span class="math-tex">\(Area\quad is\quad the\quad 6\quad faces\quad added\quad together\\ A\quad =\quad 2{ x }^{ 2 }+2xh+2xh\\ A=2{ x }^{ 2 }+4xh\)</span></p><p>b) This part is tricky but is all about making h the subject of the formula</p><p> <span class="math-tex">\(A=2{ x }^{ 2 }+4xh\\ 600=2{ x }^{ 2 }+4xh\\ 600-2{ x }^{ 2 }=4xh\\ \frac { 600-2{ x }^{ 2 } }{ 4x } =h\\ \frac { 300-{ x }^{ 2 } }{ 2x } =h\)</span></p><p>c) This stage involves substitutuing the expression for h in to the expression for volume.</p><p><span class="math-tex">\(Since\quad V{ =x }^{ 2 }h\quad and\quad h=\frac { 300-{ x }^{ 2 } }{ 2x } \\ V=\frac { { x }^{ 2 }(300-{ x }^{ 2 }) }{ 2x } \\ V=\frac { 300{ x }^{ 2 }-{ x }^{ 4 } }{ 2x } \\ (Divide\quad through\quad by\quad 2x)\\ V=150x-\frac { 1 }{ 2 } { x }^{ 3 }\)</span></p><p>d) Differentiate the function and solve for gradient = 0</p><p><span class="math-tex">\(if\quad V=150x-\frac { 1 }{ 2 } { x }^{ 3 }\quad then,\frac { dV }{ dx } =150-\frac { 3 }{ 2 } { x }^{ 2 }\\ Volume\quad is\quad a\quad maximum\quad when\quad \frac { dV }{ dx } =150-\frac { 3 }{ 2 } { x }^{ 2 }=0\\ 150=\frac { 3 }{ 2 } { x }^{ 2 }\\ 300=3{ x }^{ 2 }\\ 100={ x }^{ 2 }\\ x=\quad 10\quad (-10\quad is\quad not\quad in\quad the\quad valid\quad range)\)</span></p><p>e) This involves substituting the value of x that gives a minimum into the expression for volume.</p><p><span class="math-tex">\(Maximum\quad occurs\quad at\quad x=\quad 10\\ substitute\quad x=10\quad into\quad V=150x-\frac { 1 }{ 2 } { x }^{ 3 }\\ V=1000{ cm }^{ 3 }\)</span></p><p>f) Now substitute the value of x in to the expression for h.</p><p><span class="math-tex">\(Since\quad \frac { 300-{ x }^{ 2 } }{ 2x } =h\quad and\quad x=10\\ then,\quad \frac { 300-100 }{ 20 } =h\\ 10=h\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>Question 10</p><p>Consider a closed cylinder, radius r, height, h, whose volume must be 400<span class="math-tex">\({ cm }^{ 3 }\)</span>. The aim is to minimise the Surface area, A, required to make the cylinder.</p><p>a) if the height h, can be expressed as <span class="math-tex">\(h=\frac { a }{ \pi { r }^{ b } } \)</span>, what are the values of a and b?</p><p>b) The surface area of the cylinder can be expressed as <span class="math-tex">\(A=2\pi { r }^{ 2 }+p{ r }^{ q }\)</span>, what are the values of p and q?</p><p>c) What value of r will give the minimum surface area?</p><p>d) Use the rounded answer to the part c) to work out,</p><p>i) The value of h when A is a minimum</p><p>ii) The minimum value of A</p></div><div class="q-answer"><p>a) a = <input type="text" style="height: auto;" data-c="400"> <span class="review"></span> , b = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> </p><p>b) p = <input type="text" style="height: auto;" data-c="800"> <span class="review"></span> , q = <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span> </p><p>c) r = <input type="text" style="height: auto;" data-c="ANSWER"> <span class="review"></span> (Answer correct to 3sf)</p><p>d) i) h = <input type="text" style="height: auto;" data-c="8.00"> <span class="review"></span> cm , ii) A = <input type="text" style="height: auto;" data-c="301"> <span class="review"></span> cm<sup>2</sup> (answers to 3sf)</p></div><div class="q-explanation"><p>a) Use the formula for volume of a cylinder and rearrange.</p><p><span class="math-tex">\(V=\pi { r }^{ 2 }h\quad and\quad is\quad given\quad as\quad 400{ cm }^{ 3 }\\ then\\ 400=\pi { r }^{ 2 }h\\ and\quad so,\quad h=\frac { 400 }{ \pi { r }^{ 2 } } \)</span></p><p>b) Use the formula for surface area of a cylinder and substitute the expression for h, then simplify.</p><p><span class="math-tex">\(A=2\pi { r }^{ 2 }+2\pi { r }h\quad and\quad h=\frac { 400 }{ \pi { r }^{ 2 } } \\ Substituting\quad for\quad h,\\ A=2\pi { r }^{ 2 }+\frac { 2\pi { r }\times 400 }{ \pi { r }^{ 2 } } \\ Cancelling\quad common\quad factors\\ A=2\pi { r }^{ 2 }+\frac { 800 }{ { r } } \\ A=2\pi { r }^{ 2 }+800{ r }^{ -1 }\)</span></p><p>c) Solve for gradient = 0. Perfectly acceptable to use the solve function to do this.</p><p><span class="math-tex">\(If\quad A=2\pi { r }^{ 2 }+800{ r }^{ -1 },\quad then\quad \frac { dA }{ dr } =4\pi { r }-800{ r }^{ -2 }\\ At\quad minimum,\quad \frac { dA }{ dr } =4\pi { r }-800{ r }^{ -2 }=0\\ 4\pi { r }=800{ r }^{ -2 }\\ 4\pi { r }=\frac { 800 }{ { r }^{ 2 } } \\ 4\pi { r }^{ 3 }=800\\ { r }^{ 3 }=\frac { 800 }{ 4\pi } \\ r=\sqrt [ 3 ]{ \frac { 800 }{ 4\pi } } \\ r=3.9929454...\\ r=3.99\quad cm\quad (3sf)\)</span></p><p>d) Substitutions can be done using the table function on your GDC or otherwise.</p><p>i) <span class="math-tex">\(Substitute\quad r=3.99\quad in\quad to\quad h=\frac { 400 }{ \pi { r }^{ 2 } } \\ h=7.997685...\\ h=8.00cm\quad (3sf)\)</span></p><p>ii) <span class="math-tex">\(Substitute\quad r=3.99\quad in\quad to\quad A=2\pi { r }^{ 2 }+800{ r }^{ -1 }\\ A=300.5301915...\\ A=\quad 301{ cm }^{ 2 }\quad (3sf)\)</span></p></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="purple"> <h3>Revision Cards</h3> <p>We reconginse that when preparing for assessments, that you will want shorter and shorter references. For that reason, each topic comes with a short set of revision cards for really quick reference.</p> <div class="panel panel-has-footer panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Example Revision Cards</p> </div> </div> <div class="panel-body"> <div> <p>Here is an example of a quick review of culmulative frequency and box plots.</p> <div id="carousel-236" class="dynamic-gallery carousel slide" data-id="236"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-236/cfbprc001.jpeg" data-fancybox="gallery-236" title="" data-caption=""><img alt="" src="../../../std-galleries/15-236/cfbprc001.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-236/cfbprc002.jpeg" data-fancybox="gallery-236" title="" data-caption=""><img alt="" src="../../../std-galleries/15-236/cfbprc002.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-236/cfbprc003.jpeg" data-fancybox="gallery-236" title="" data-caption=""><img alt="" src="../../../std-galleries/15-236/cfbprc003.jpeg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-236/cfbprc004.jpeg" data-fancybox="gallery-236" title="" data-caption=""><img alt="" src="../../../std-galleries/15-236/cfbprc004.jpeg"></a></div></div><a class="left carousel-control" href="#carousel-236" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-236" 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/15-236/cfbprc001-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-236/cfbprc002-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-236/cfbprc003-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-236/cfbprc004-thumb128.jpg"><li></li></ol> </div> </div> <div class="panel-footer"> </div> </div> </div> <div class="greyBg"> <h3>Exam Style Questions</h3> <p>The end of each chapter has a selection of questions like the ones you can expect in your assessments. You can download and print a version of these from the top of the topic page and then watch video solutions! 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