A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 53 cells. when will the population reach 20,000 using exponential growth 

To determine when the population of **Escherichia coli** will reach 20,000 cells, we can use the exponential growth formula:

\[
P(t) = P_0 \cdot e^{rt}
\]

where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate,
- \( t \) is the time.

Since the bacteria double every 20 minutes, we know that after 20 minutes, \( P(20) = 2P_0 \). This gives us a way to calculate the growth rate \( r \).

1. **Determine the growth rate \( r \)**:
   \[
   2P_0 = P_0 \cdot e^{r \cdot 20}
   \]
   Dividing both sides by \( P_0 \):
   \[
   2 = e^{r \cdot 20}
   \]
   Taking the natural logarithm of both sides:
   \[
   \ln(2) = r \cdot 20
   \]
   Solving for \( r \):
   \[
   r = \frac{\ln(2)}{20}
   \]

2. **Substitute the known values**:
   - Initial population, \( P_0 = 53 \),
   - Desired population, \( P(t) = 20,000 \),
   - Growth rate \( r = \frac{\ln(2)}{20} \).

   We want to find \( t \) when \( P(t) = 20,000 \):
   \[
   20,000 = 53 \cdot e^{rt}
   \]
   Solving for \( t \):
   \[
   \frac{20,000}{53} = e^{rt}
   \]
   Taking the natural logarithm of both sides:
   \[
   \ln\left(\frac{20,000}{53}\right) = rt
   \]
   Substituting \( r \) into the equation:
   \[
   t = \frac{\ln\left(\frac{20,000}{53}\right)}{\frac{\ln(2)}{20}}
   \]

Now let's calculate the exact value of \( t \).The population of **Escherichia coli** will reach 20,000 cells in approximately **171.2 minutes** (or about 2 hours and 51 minutes) under the given conditions.

Would you like any further details or have any questions?

Here are some related questions to explore:
1. How would the time change if the bacteria doubled every 15 minutes instead of 20 minutes?
2. What would be the population after 2 hours?
3. How does the exponential growth model compare to a logistic growth model?
4. If the initial population was 100 cells, how long would it take to reach 20,000 cells?
5. What are the assumptions of the exponential growth model, and when might they not hold true?

**Tip:** Exponential growth can quickly lead to large numbers, making it essential to understand the conditions under which the model is accurate.

Learn how to calculate population growth using the exponential growth formula with a detailed example. Find out when the population of Escherichia coli will reach 20,000 cells starting from an initial population of 53 cells, doubling every 20 minutes. Explore related questions on growth rate and time calculations.

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