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61 lines
3 KiB
HTML
61 lines
3 KiB
HTML
<div class="book-prose">
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<h2 id="history" class="book-prose__heading">History</h2>
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<p>
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Variation models grew from early physics and astronomy, where scientists needed formulas for how
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one quantity depends on another. The language of proportionality became a bridge between algebra
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and real measurements.
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</p>
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<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
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<h3 id="why-constant-k" class="book-prose__heading">Why we use a constant \(k\)</h3>
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<p>
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“Directly proportional” means ratio \(\frac{y}{x}\) stays constant, so \(\frac{y}{x}=k\) and
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\(y=kx\). For inverse variation, product \(xy\) stays constant, so \(xy=k\).
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</p>
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<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
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<details class="book-question">
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<summary class="book-question__prompt">If \(y\propto x\) and \(y=15\) when \(x=3\), find \(y\) when \(x=10\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(k=5\Rightarrow y=5x\Rightarrow y=50\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">If \(y\propto \frac1x\) and \(y=8\) when \(x=5\), find \(y\) when \(x=20\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(k=40\Rightarrow y=\frac{40}{x}\Rightarrow y=2\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">\(y\) varies as \(x^2\). If \(y=27\) when \(x=3\), find \(y\) when \(x=5\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(y=kx^2,\ k=3\Rightarrow y=3(25)=75\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">\(y\) varies jointly as \(x\) and \(z\). If \(y=30\) when \(x=3,z=2\), find \(y\) when \(x=5,z=4\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(y=kxz,\ k=5\Rightarrow y=5(5)(4)=100\).</p>
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</div>
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</details>
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<details class="book-question">
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<summary class="book-question__prompt">A taxi fare \(F\) varies partly as distance \(d\): \(F=md+c\). If \(F=26\) at \(d=8\), and \(F=38\) at \(d=14\), find \(m,c\).</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> Solve two equations: \(8m+c=26,\ 14m+c=38\).</p>
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<p>\(6m=12\Rightarrow m=2,\ c=10\). So \(F=2d+10\).</p>
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</div>
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</details>
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<h2 id="applications" class="book-prose__heading">Applications</h2>
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<ul>
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<li><strong>Physics:</strong> pressure, speed, and force models often use direct/inverse variation.</li>
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<li><strong>Business:</strong> cost models combine fixed and variable components.</li>
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<li><strong>Engineering:</strong> scaling laws use power variation relations.</li>
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</ul>
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<h2 id="references" class="book-prose__heading">References</h2>
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<ul>
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<li>Standard high-school algebra units on direct and inverse variation.</li>
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<li>Elementary modeling texts for joint and partial variation.</li>
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</ul>
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</div>
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