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<title>Adventure 10 — Activity</title>
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<a href="index.html"><button>🧭 Hub</button></a>
<a href="adventure_10_home.html"><button>🏠 Home</button></a>
<a href="adventure_10_solution.html"><button>✅ Solutions</button></a>
<a href="adventure_10_charts.pdf" target="_blank" rel="noopener"><button class="secondary">📈 Charts PDF</button></a>
</div>
<h1>✏️ Adventure 10 — Activity</h1>
<div class="subtitle">Feynman–Gamow Mission — ratios, DiVA, and the long-term idea behind L’Hôpital.</div>
<div class="card"><p class="subtitle" style="margin:0;">
Use the chart to estimate values and compare the two probe distances. Close estimates are fine. Units: <b>t</b> in hours, distances in mission distance units.
</p>
</div>
</div>
<div class="card" style="margin-top:1.2rem;">
<div style="text-align:center;">
<img
src="images/adventure_10_chart_full.png"
alt="Feynman–Gamow Mission DiVA Charts"
style="width: 80%; max-width: 1020px; height: auto; border-radius: 14px; border: 1px solid #dde3ff; background: #fff;"
>
<div style="font-size:0.95rem; color:#666; margin-top:0.4rem;">
Feynman–Gamow Mission — distance, velocity, and acceleration comparisons
</div>
</div>
</div>
<div class="content-shift">
<h2>🧩 Part 1 — Mission rule</h2>
<div class="lookback">
<span class="dot green"></span>
Look back at the distance curves above
</div>
<p>The mission requires Feynman to stay at least twice as far from Earth as Gamow.</p>
<div class="row-grid">
<div>
<label for="rule"><b>1A)</b> Write the mission rule using a ratio</label>
<input id="rule" type="text" placeholder="Example: f(t)/g(t) >= x">
</div>
<div>
<label for="meaning"><b>1B)</b> What does this rule mean in words?</label>
<input id="meaning" type="text" placeholder="Explain in one sentence">
</div>
</div>
<h2>🧮 Part 2 — Derivatives</h2>
<p>Use the chart and formulas to identify the velocity and acceleration functions.</p>
<div class="row-grid">
<div>
<label for="fp"><b>2A)</b> What is <code>f'(t)</code>?</label>
<input id="fp" type="text" placeholder="">
</div>
<div>
<label for="gp"><b>2B)</b> What is <code>g'(t)</code>?</label>
<input id="gp" type="text" placeholder="">
</div>
<div>
<label for="fpp"><b>2C)</b> What is <code>f''(t)</code>?</label>
<input id="fpp" type="text" placeholder="">
</div>
<div>
<label for="gpp"><b>2D)</b> What is <code>g''(t)</code>?</label>
<input id="gpp" type="text" placeholder="">
</div>
</div>
<h2>📊 Part 3 — Distance ratios at marked times</h2>
<div class="lookback">
<span class="dot green"></span>
Use the distance chart above
</div>
<p>Fill in the table from the chart or by calculation. Then check your answers.</p>
<table>
<tr><th>t (hours)</th><th>f(t)</th><th>g(t)</th><th>f(t)/g(t)</th></tr>
<tr>
<td>5</td>
<td><input id="f5" class="table-input" type="number" step="1"></td>
<td><input id="g5" class="table-input" type="number" step="1"></td>
<td><input id="r5" class="table-input" type="number" step="0.01"></td>
</tr>
<tr>
<td>10</td>
<td><input id="f10" class="table-input" type="number" step="1"></td>
<td><input id="g10" class="table-input" type="number" step="1"></td>
<td><input id="r10" class="table-input" type="number" step="0.01"></td>
</tr>
<tr>
<td>20</td>
<td><input id="f20" class="table-input" type="number" step="1"></td>
<td><input id="g20" class="table-input" type="number" step="1"></td>
<td><input id="r20" class="table-input" type="number" step="0.01"></td>
</tr>
<tr>
<td>30</td>
<td><input id="f30" class="table-input" type="number" step="1"></td>
<td><input id="g30" class="table-input" type="number" step="1"></td>
<td><input id="r30" class="table-input" type="number" step="0.01"></td>
</tr>
<tr>
<td>50</td>
<td><input id="f50" class="table-input" type="number" step="1"></td>
<td><input id="g50" class="table-input" type="number" step="1"></td>
<td><input id="r50" class="table-input" type="number" step="0.01"></td>
</tr>
</table>
<h2>🚀 Part 4 — Long-time ratios</h2>
<div class="lookback">
<span class="dot red"></span><span class="dot orange"></span>
Use f(t)=2t²+175t f'(t)=4t and g(t)=t²+5t g'(t)=2 to Fill in the distance and velocity ratio tables.
</div>
<p>What do you notice?</p>
<table>
<tr><th>t (hours)</th><th>Distance ratio f(t)/g(t)</th><th>Velocity ratio f'(t)/g'(t)</th></tr>
<tr>
<td>240</td>
<td><input id="lr240" class="table-wide" type="number" step="0.01"></td>
<td><input id="vr240" class="table-wide" type="number" step="0.01"></td>
</tr>
<tr>
<td>720</td>
<td><input id="lr720" class="table-wide" type="number" step="0.01"></td>
<td><input id="vr720" class="table-wide" type="number" step="0.01"></td>
</tr>
<tr>
<td>1200</td>
<td><input id="lr1200" class="table-wide" type="number" step="0.01"></td>
<td><input id="vr1200" class="table-wide" type="number" step="0.01"></td>
</tr>
</table>
<div class="row-grid">
<div><label for="accRatio"><b>4A)</b> What is <code>f''(t)/g''(t)</code>?</label><input id="accRatio" type="text" placeholder=""></div>
<div><label for="limitRatio"><b>4B)</b> As <code>t → ∞</code>, what value do the distance and velocity ratios approach?</label><input id="limitRatio" type="text" placeholder=""></div>
</div>
<h2>🧠 Part 5 — Explain the chain of reasoning</h2>
<p>Use one or two sentences to explain why the distance ratio approaches 2.</p>
<div class="row-grid">
<div>
<label for="explain"><b>5A)</b> Explanation</label>
<textarea id="explain" rows="4" placeholder="Mention acceleration ratio, then velocity ratio, then distance ratio."></textarea>
</div>
<div>
<label for="challenge"><b>5B)</b> Challenge: for same-degree polynomials, what does the ratio approach?</label>
<input id="challenge" type="text" placeholder="Example: ratio of leading coefficients">
</div>
</div>
</div>
<div class="page">
<div class="buttons" style="display:flex; flex-wrap:wrap; gap:12px; margin-bottom:0.8rem;">
<button type="button" onclick="checkAnswers()" id="checkBtn">✅ Check my answers</button>
<button type="button" class="secondary" onclick="fillSample()" id="sampleBtn">Use sample values</button>
<button type="button" class="secondary" onclick="toggleHint()">💡 Show / hide hint</button>
<button type="button" class="secondary" onclick="toggleSolution()">📄 Show / hide solution</button>
</div>
</div>
<div id="hintBox" class="hint">
<ul style="margin:0.2rem 0 0.2rem 1.2rem;">
<li>The mission rule is a ratio statement: Feynman must stay at least twice as far from Earth as Gamow.</li>
<li>The second derivatives are constants, so their ratio is the easiest place to start.</li>
<li>The first table values are 925/50, 1950/150, 4300/500, 7050/1050, and 13750/2750.</li>
<li>The long-time table shows the same pattern in both distance and velocity ratios: both approach 2.</li>
<li>At long times, same-degree polynomials are controlled by their leading terms.</li>
</ul>
</div>
<div id="solutionBox" class="solution">
<strong>Sample answers (for checking)</strong>
<ul>
<li><code>f(t)/g(t) ≥ 2</code> for all <code>t ≥ 0</code>.</li>
<li><code>f'(t)=4t+175</code>, <code>g'(t)=2t+5</code>, <code>f''(t)=4</code>, <code>g''(t)=2</code>.</li>
<li>Table 1: (925, 50, 18.50), (1950, 150, 13.00), (4300, 500, 8.60), (7050, 1050, 6.71), (13750, 2750, 5.00).</li>
<li>Table 2: (2.67, 2.34), (2.23, 2.11), (2.14, 2.07).</li>
<li><code>f''(t)/g''(t)=2</code>, and both long-time ratios approach <code>2</code>.</li>
<li>For same-degree polynomials, the ratio approaches the ratio of the leading coefficients.</li>
</ul>
</div>
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