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59 lines
2.8 KiB
HTML
59 lines
2.8 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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Mensuration developed from practical needs in architecture, agriculture, and trade. Ancient
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mathematicians studied area and volume formulas, with later refinement by Archimedes and others.
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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-one-third" class="book-prose__heading">Why pyramid and cone have one-third factor</h3>
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<p>
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A pyramid (or cone) with same base area and height as a prism (or cylinder) has one-third the
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volume. This can be shown by dissection arguments or calculus in advanced courses.
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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">Find volume of a cylinder with \(r=3\) cm, \(h=10\) cm.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(V=\pi r^2h=90\pi\) cm\(^3\).</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">Find total surface area of a closed cylinder with \(r=2\) cm, \(h=7\) cm.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(2\pi r^2+2\pi rh=8\pi+28\pi=36\pi\) cm\(^2\).</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">Find volume of a cone with \(r=6\) cm, \(h=9\) cm.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(\frac13\pi r^2h=\frac13\pi(36)(9)=108\pi\) cm\(^3\).</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">Sphere has radius 5 cm. Find its volume.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> \(V=\frac43\pi(5^3)=\frac{500}{3}\pi\) cm\(^3\).</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 block is \(4\times3\times2\) m. Density is \(2500\) kg/m\(^3\). Find mass.</summary>
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<div class="book-question__solution">
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<p><strong>Answer:</strong> Volume \(=24\) m\(^3\). Mass \(=2500\times24=60000\) kg.</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>Construction:</strong> concrete volume and paint area estimation.</li>
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<li><strong>Packaging:</strong> minimizing material for fixed volume containers.</li>
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<li><strong>Engineering:</strong> tank capacity and material mass calculations.</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 mensuration units on 3D solids.</li>
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<li>Geometry texts on prism, cylinder, cone, pyramid, and sphere formulas.</li>
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</ul>
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</div>
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