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<h1>✅ Adventure 9 — Solutions</h1>
<div class="subtitle">These solutions match the Student Activity prompts and use the DiVA chart for estimates.</div>

<div class="chart-float">
  <img src="images/adventure_9_chart.png" alt="Adventure 9 DiVA Charts (U-235)">
  <div style="font-size:0.9rem; color:#666; margin-top:0.35rem;">U‑235 DiVA Chart (L, M, v, a)</div>
</div>

<div class="clearfix">
<h2>🧩 Part 1 — Derivatives</h2>
<ul>
<li><code>d/dx(e^x) = e^x</code></li>
<li><code>d/dx(e^{g(x)}) = g'(x)e^{g(x)}</code></li>
</ul>

<h2>🧱 Part 2 — Decay constant</h2>
<ul>
<li>Half‑life condition: <code>(1/2)M(0) = M(0)e^{-kT}</code></li>
<li>Cancel <code>M(0)</code>: <code>1/2 = e^{-kT}</code></li>
<li>Take ln: <code>ln(1/2) = -kT</code></li>
<li>So <code>k = ln(2)/T</code>. With <code>T=0.704</code>, <code>k≈0.9846</code>.</li>
</ul>

<h2>📊 Part 3 — Half‑life table</h2>
<div class="card"><ul style="margin:0.2rem 0 0.2rem 1.2rem;">
<li>1024 → 512 → 256 → 128 → 64 → 32 → 16 → 8</li>
<li>11 g is between 16 and 8 → about <b>6.5 half‑lives</b></li>

</ul></div>
<h2>📊 Part 4 — Age of the Earth</h2>
<div class="card"><ul style="margin:0.2rem 0 0.2rem 1.2rem;">

<li>Age ≈ <code>6.5 × 0.704 = 4.58</code> billion years</li>
</ul></div>
<h2>🧮 Part 5 — Chart estimates (example)</h2>
<ul>
<li><code>L(1) ≈ 644</code> g</li>
<li><code>L(2) ≈ 884</code> g</li>
<li><code>M(1) ≈ 1024 − 644 ≈ 380</code> g</li>
</ul>

<h2>🧠 Part 6 — Integral meaning</h2>
<p><code>∫₀¹ v(t) dt</code> is the total amount of uranium that disappeared (and became lead) from 0 to 1 billion years — i.e., the increase in <code>L(t)</code> over that interval L(1)=644.</p>
<p>M(1)= Uranium remaining from chart 380 grams - Notice M(1) + L(1) = M(0) = 1024 grams, so the area under v(t) from 0 to 1 must be L(1) = 644 grams.</p>



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