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This question describes exponential growth of a bacterium culture. Let me guide you through each part and provide detailed solutions.

### Problem Data:
- Initial population (\( P_0 \)) = 50 cells
- Population after 1.5 hours = 875 cells
- The growth is proportional to the population, meaning the equation follows the form of **exponential growth**:
  \[
  P(t) = P_0 e^{kt}
  \]
Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the growth rate constant,
- \( t \) is time in hours.

### (a) **Finding the expression for \( P(t) \):**
We know the population grows exponentially, so:
\[
P(t) = 50 e^{kt}
\]
To find \( k \), we use the fact that after 1.5 hours, the population is 875:
\[
875 = 50 e^{1.5k}
\]
Solve for \( k \):
\[
\frac{875}{50} = e^{1.5k} \quad \Rightarrow \quad 17.5 = e^{1.5k}
\]
Take the natural logarithm of both sides:
\[
\ln(17.5) = 1.5k \quad \Rightarrow \quad k = \frac{\ln(17.5)}{1.5}
\]
Using a calculator:
\[
k \approx \

A culture of the bacterium Salmonella enteritidis initially contains 50 cells. When introduced into a nutrient broth, the culture grows at a rate proportional to its size. After 1.5 hours, the population has increased to 875. (a) Find an expression for the number of bacteria after t hours. (b) Find the number of bacteria after 5 hours. (c) Find the rate of growth after 5 hours. (d) After how many hours will the population reach 250,000?

Learn how to solve an exponential growth problem for a bacterium culture with step-by-step solutions. Topics include finding the population at a given time, determining the growth rate, and calculating the time required for a population to reach a specific size. Suitable for college-level calculus courses.

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