Exponential Growth and Decay Models

1. 1 A bacterium doubles every hour. Start: 1. After 1h: 2. After 2h: 4. After 3h: 8. Compare with a colony growing by 10 per hour (linear). Plot both to n=8. 'Exponential growth eventually outpaces any linear growth — dramatically.'
2. 2 N = N₀ × aᵗ where N₀ = initial amount, a = growth factor (a>1 for growth, 0<a<1 for decay), t = time. For the bacteria: N = 1 × 2ᵗ. For 5% annual growth from initial 1000: N = 1000 × 1.05ᵗ. Students identify N₀ and a in three models.
3. 3 A radioactive substance halves every 10 years (half-life = 10). N = N₀ × (0.5)^(t/10). Or equivalently: N = N₀ × e^(−λt) where λ = ln2/10. Calculate remaining amount after 30 years from 200g: N = 200 × (0.5)³ = 25g. Students apply to three decay problems.
4. 4 When does 1000 × 1.05ᵗ reach 2000? 1.05ᵗ = 2. Take log: t log 1.05 = log 2. t = log2/log1.05 ≈ 14.2 years. Students solve four 'when does it reach?' problems using logs.
5. 5 A cooling model: T = 20 + 60e^(−0.1t). 'What is the initial temperature?' (20+60=80°C.) 'What does the 20 represent?' (Room temperature — the long-term value.) 'What does 0.1 represent?' (Rate of cooling.) Students interpret two more models.
6. 6 A population is modelled as P = 1000 × 1.02ᵗ. 'Predict population in 200 years.' Calculate. 'Does this seem realistic? What assumptions does the model make? What real factors might it ignore?' Discuss limitations.
7. 7 If y = aᵇˣ, then log y = log a + x log b — a linear relationship. Students plot log y against x for exponential data and observe a straight line, connecting exponential models to linear regression.

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