The model is N(t) = 1000(1.3)^t, where t is hours since the start.
- Initial amount: N(0) = 1000(1.3)^0 = 1000 bacteria.
- Hourly change: Each hour the population is multiplied by 1.3, i.e. it increases by 30% every hour.
- Examples:
- N(1) = 1000·1.3 = 1300
- N(2) = 1000·1.3^2 = 1690
- N(3) = 1000·1.3^3 = 2197
- N(4) = 1000·1.3^4 ≈ 2856.1
- Doubling time: Solve 1000(1.3)^t = 2000 ⇒ (1.3)^t = 2 ⇒ t = ln(2)/ln(1.3) ≈ 2.64 hours. So the population roughly doubles every 2.64 hours.
- Instantaneous (continuous) growth rate: dN/dt = 1000(1.3)^t·ln(1.3) = N(t)·ln(1.3). Since ln(1.3) ≈ 0.2624, the instantaneous growth is about 0.2624·N(t) bacteria per hour (≈26.24% per hour on a continuous scale).
- Long-term behavior and assumptions: The model predicts exponential growth without bound as t increases. This is an idealized model that assumes unlimited resources and no other limiting factors; in real experiments growth usually slows once resources become limited.
Summary: According to the model, the bacteria population grows exponentially: it starts at 1000 and increases by 30% each hour (multiplying by 1.3 every hour), roughly doubling every 2.64 hours.