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The question involves an exponential growth problem. Let's break it down and solve both parts:

### Given:
- Initial population \( P_0 = 265 \) million
- Growth rate \( r = 4.2\% = 0.042 \) per hour
- Time for first part: \( t = 3 \) hours

### Formula for exponential growth:
\[
P(t) = P_0 \times e^{r t}
\]
Where:
- \( P(t) \) is the population at time \( t \)
- \( P_0 \) is the initial population
- \( r \) is the growth rate
- \( t \) is time in hours
- \( e \) is the base of the natural logarithm, approximately 2.71828

---

### 1.5.1 Calculate the population after 3 hours

Substitute the values into the exponential growth formula:
\[
P(3) = 265 \times e^{0.042 \times 3}
\]

I'll calculate this next.The population of microbes in the petri dish after 3 hours will be approximately **300.58 million**.

---

### 1.5.2 How long will it take for the population to triple to 795 million?

We are asked to find the time \( t \) when the population reaches 795 million, given an initial population of 265 million. The formula is the same:

\[
P(t) = P_0 \times e^{r t}
\]

In this case:
- \( P(t) = 795 \) million
- \( P_0 = 265 \) million
- \( r = 0.042 \)

Rearrange the equation to solve for \( t \):
\[
t = \frac{\ln\left(\frac{P(t)}{P_0}\right)}{r}
\]

Substituting the values:
\[
t = \frac{\ln\left(\frac{795}{265}\right)}{0.042}
\]

I'll now calculate this.It will take approximately **26.16 hours** for the population to triple to 795 million.

---

Would you like further clarification or details on any part of the process? Here are some related questions:

1. How does the exponential growth formula work in biological systems?
2. What is the significance of the base \( e \) in exponential growth?
3. How would the result change if the growth rate were different?
4. Can exponential growth continue indefinitely in a real-life scenario?
5. How would you adjust the formula for different initial populations?

**Tip:** Exponential growth describes many natural phenomena, but real-world constraints like resource limits often slow down the growth over time.

A medical sample contains 265 million microbes in a petri dish, with an exponential population growth rate of 4.2% per hour. Calculate the population of microbes in the petri dish after 3 hours. How long will it take for the population to triple to 795 million?

Learn how to calculate the exponential growth of microbes in a petri dish, given an initial population of 265 million and a growth rate of 4.2% per hour. Find out how long it takes for the population to triple to 795 million. Suitable for students in Grades 9-12.

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