✅ Adventure 9 — Solutions
These solutions match the Student Activity prompts and use the DiVA chart for estimates.
U‑235 DiVA Chart (L, M, v, a)
🧩 Part 1 — Derivatives
d/dx(e^x) = e^xd/dx(e^{g(x)}) = g'(x)e^{g(x)}
🧱 Part 2 — Decay constant
- Half‑life condition:
(1/2)M(0) = M(0)e^{-kT} - Cancel
M(0):1/2 = e^{-kT} - Take ln:
ln(1/2) = -kT - So
k = ln(2)/T. WithT=0.704,k≈0.9846.
📊 Part 3 — Half‑life table
- 1024 → 512 → 256 → 128 → 64 → 32 → 16 → 8
- 11 g is between 16 and 8 → about 6.5 half‑lives
📊 Part 4 — Age of the Earth
- Age ≈
6.5 × 0.704 = 4.58billion years
🧮 Part 5 — Chart estimates (example)
L(1) ≈ 644gL(2) ≈ 884gM(1) ≈ 1024 − 644 ≈ 380g
🧠 Part 6 — Integral meaning
∫₀¹ v(t) dt is the total amount of uranium that disappeared (and became lead) from 0 to 1 billion years — i.e., the increase in L(t) over that interval L(1)=644.
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.