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137 lines
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HTML
137 lines
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<h1 align="center">Mathematical Models 2</h1>
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<p align="center"><i>How to translate from English to Mathematics</i> </p>
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<h2>The Real World</h2>
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<p>Most mathematics questions are neat and pure and simple. </p>
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<p>But the real world is often "messy": there are facts that get in the way, the numbers don't always work out nicely, we have to convert answers to different units and so on.</p>
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<p>On <a href="mathematical-models.html">Mathematical Models</a> we showed how to make and use some simple models. Here are some more.</p>
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<p> </p>
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<h2>Example: River Cruise</h2>
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<h3>Example: a 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey? </h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/river-boat.svg" alt="river boat back-and-forth in 3 hours" /></p>
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<p>There are two speeds to think about: the speed the boat makes in the water, and the speed relative to the land:</p>
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<ul>
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<li>Let <b>b</b> = the boat's speed in the water (km/h)</li>
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<li>Let <b>v</b> = the speed relative to the land (km/h)</li>
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</ul>
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<p>Because the river flows downstream at 2 km/h:</p>
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<ul>
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<li>when going upstream, <b>v = b-2</b> (its speed is reduced by 2 km/h)</li>
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<li>when going downstream, <b>v = b+2</b> (its speed is increased by 2 km/h)</li>
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</ul>
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<p>We can turn those speeds into times using:</p>
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<p align="center" class="larger">time = distance / speed</p>
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<p align="center">(moving 8 km at 4 km/h takes 8/4 = 2 hours, right?)</p>
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<p>And we know the total time is 3 hours:</p>
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<p align="center" class="larger">total time = time upstream + time downstream = 3 hours</p>
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<p>Put all that together:</p>
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<p align="center"><span class="larger">total time = 15/(b-2) + 15/(b+2) = 3 hours</span></p>
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<p>Now we use our algebra skills to solve for "b".</p>
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<p>First, get rid of the fractions by multiply through by <b>(b-2)</b><b>(b+2)</b>: </p>
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<p align="center" class="larger">3(b-2)(b+2) = 15(b+2) + 15(b-2)</p>
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<p>Expand everything:</p>
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<p align="center" class="larger">3(b<sup>2</sup>-4)= 15b+30 + 15b-30</p>
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<p>Bring everything to the left and simplify:<b></b></p>
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<p align="center" class="larger">3b<sup>2</sup> - 30b -12 = 0</p>
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<p>It is a <a href="quadratic-equation.html">Quadratic Equation</a>! And <a href="../quadratic-equation-solver.html">solving it</a> gives the solutions:</p>
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<p align="center"><span class="large">b = -0.39 or 10.39 </span>(to 2 decimal places)</p>
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<p>b =-0.39 makes no sense for this real world question, but b = 10.39 is just perfect!</p>
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<p align="center">Answer: <span class="large">Boat's Speed = 10.39 km/h </span>(to 2 decimal places)</p>
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<p align="center">And so the upstream journey = 15 / (10.39-2) = 1.79 hours = <span class="large">1 hour 47min</span></p>
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<p align="center">And the downstream journey = 15 / (10.39+2) = 1.21 hours = <span class="large">1 hour 13min</span></p>
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<p> </p>
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<h2>Example: Mixing Solutions</h2>
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<h3>You have been asked to provide 150 mL of hydrochloric acid at a 12% concentration</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/chemicals.jpg" width="120" height="161" alt="chemicals" /></p>
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<p>In the cabinet you find:</p>
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<ul>
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<li>one bottle marked <b>10% HCl</b> (10% hydrochloric acid)</li>
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<li>and another bottle marked <b>25% HCl</b> (25% hydrochloric acid)</li>
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</ul>
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<p>How many mL of each should you use?</p>
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<ul>
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<li>a = quantity of 10% solution</li>
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<li>150 - a = quantity of 25% solution</li>
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</ul>
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<p align="left">total HCl = HCl in 10% solution + HCl in 25% solution</p>
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<div class="so">(0.12)(150) = (0.10)(a) + (0.25)(150-a)</div>
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<p align="left">Solve: 18 = 0.1a + 37.5 - 0.25a</p>
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<p align="left">"a"s to left, rest to right: 0.25a - 0.1a = 37.5-18</p>
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<p align="left">Simplify: 0.15a = 19.5</p>
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<div class="so">a = 19.5/0.15 = 130</div>
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<div class="center80">
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<p align="center">Answer: <b>130 ml of 10%</b> solution and <b>20 ml of 25%</b> solution</p>
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</div>
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<p align="center">Note: I have a "<a href="../numbers/mix-slider.html">Mix Slider</a>" that could also have been used to solve this</p>
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<p> </p>
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<p> </p>
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<div class="related"><a href="mathematical-models.html">Mathematical Models</a><a href="index.html">Algebra Index</a></div>
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