The logistic growth model Upper P left parenthesis t right parenthesis equals StartFraction 3000 Over 1 plus 33.03 e Superscript negative 0.455 t EndFraction
 represents the population​ (in grams) of a bacterium after t hours. Answer parts​ (a) through​ (e).
What is the growth rate of the​ bacteria?

The given logistic growth model is:

\[
P(t) = \frac{3000}{1 + 33.03 e^{-0.455t}}
\]

This represents the population of a bacterium (in grams) after \(t\) hours.

To find the growth rate of the bacteria, we need to focus on the coefficient of \(t\) in the exponential term. The logistic growth model can generally be written in the form:

\[
P(t) = \frac{L}{1 + A e^{-rt}}
\]

where:
- \(L\) is the carrying capacity (maximum population, in this case, 3000 grams),
- \(r\) is the growth rate of the bacteria,
- \(A\) is a constant, and
- \(t\) is time.

In the given model:
- \(L = 3000\),
- \(A = 33.03\), and
- \(r = 0.455\) (since the exponent is \(-0.455t\)).

Therefore, the growth rate \(r\) of the bacteria is:

\[
\boxed{0.455}
\]

This means the bacteria grow at a rate of 0.455 per hour.

Would you like to dive deeper into how this rate affects the population over time?

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Here are 5 related questions to further expand this concept:
1. What is the carrying capacity of the bacterium population?
2. How do you interpret the initial population from this model?
3. At what time does the population reach half of the carrying capacity?
4. How does changing the growth rate \(r\) affect the population curve?
5. What happens to the population as \(t \to \infty\)?

**Tip**: The logistic growth rate, \(r\), determines how quickly the population approaches its carrying capacity. A higher \(r\) results in a faster approach.

The logistic growth model P(t) = 3000 / (1 + 33.03e^(-0.455t)) represents the population​ (in grams) of a bacterium after t hours. What is the growth rate of the​ bacteria?

Learn how to calculate the growth rate of bacteria using the logistic growth model P(t) = 3000 / (1 + 33.03e^(-0.455t)). This problem demonstrates key concepts in exponential functions and population dynamics, making it suitable for high school students (Grades 10-12).

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