The bacteria culture's growth can be modeled by an exponential growth equation:

$P(t) = P_0 \cdot 2^{t/T}$

Where:

• $P_0 = 2000$ (initial population),
• $T = 0.5$ hours (doubling time),
• $t$ is the time in hours.

---

1. Bacterial Population after 20 minutes:

Convert 20 minutes to hours: $20 \, \text{minutes} = \frac{20}{60} = \frac{1}{3} \, \text{hours}$.

Substitute into the formula: $P\left(\frac{1}{3}\right) = 2000 \cdot 2^{\frac{\frac{1}{3}}{0.5}} = 2000 \cdot 2^{\frac{2}{3}}.$

Use $2^{\frac{2}{3}} \approx 1.5874$: $P\left(\frac{1}{3}\right) \approx 2000 \cdot 1.5874 = 3174.8.$

The population after 20 minutes is approximately 3175 bacteria.

---

2. Bacterial Population after 10 hours:

Substitute $t = 10$ into the formula: $P(10) = 2000 \cdot 2^{\frac{10}{0.5}} = 2000 \cdot 2^{20}.$

Calculate $2^{20} = 1,048,576$: $P(10) = 2000 \cdot 1,048,576 = 2,097,152,000.$

The population after 10 hours is 2,097,152,000 bacteria.

---

Would you like further details? Here are 5 related questions to delve deeper:

1. How does the doubling time affect the population growth rate?
2. What is the general formula for exponential growth?
3. How do you calculate population sizes for arbitrary times not aligned with the doubling interval?
4. What happens to the population size as time approaches infinity?
5. Can this growth be modeled logarithmically to reverse engineer the time?

Tip: Always check units of time when substituting into growth formulas!