A bacteria culture starts with 800 bacteria and grows at a rate proportional to its size. After 5 hours, there are 9000 bacteria. 1. Find an expression for the number of bacteria after t hours. 2. Find the number of bacteria after 6 hours. 3. Find the growth rate after 6 hours. 4. After how many hours will the population reach 30000?

Given the problem, the bacteria culture follows **exponential growth**, where the population grows at a rate proportional to its size. This can be modeled using the differential equation:

\[
\frac{dP}{dt} = kP
\]
where:
- \( P(t) \) is the population at time \( t \),
- \( k \) is the growth constant,
- \( P_0 \) is the initial population (800 bacteria).

### Step 1: Expression for the number of bacteria after \( t \) hours

The solution to the differential equation is:

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

Using the initial condition \( P_0 = 800 \), the equation becomes:

\[
P(t) = 800 e^{kt}
\]

Now, we use the given information that after 5 hours, there are 9000 bacteria:

\[
9000 = 800 e^{5k}
\]

Solving for \( k \):

\[
e^{5k} = \frac{9000}{800} = 11.25
\]
\[
5k = \ln(11.25)
\]
\[
k = \frac{\ln(11.25)}{5} \approx 0.5057
\]

Thus, the expression for the number of bacteria after \( t \) hours is:

\[
P(t) = 800 e^{0.5057t}
\]

### Step 2: Number of bacteria after 6 hours

Substitute \( t = 6 \) into the expression:

\[
P(6) = 800 e^{0.5057 \times 6}
\]
\[
P(6) \approx 800 e^{3.0342} \approx 800 \times 20.785 \approx 16628
\]

So, the number of bacteria after 6 hours is approximately **16,628**.

### Step 3: Growth rate after 6 hours

The growth rate is given by \( \frac{dP}{dt} = kP(t) \). Using \( k = 0.5057 \) and \( P(6) \approx 16628 \):

\[
\frac{dP}{dt} \bigg|_{t=6} = 0.5057 \times 16628 \approx 8410 \, \text{bacteria/hour}
\]

So, the growth rate after 6 hours is approximately **8410 bacteria per hour**.

### Step 4: Time to reach 30,000 bacteria

We want to find \( t \) when \( P(t) = 30000 \):

\[
30000 = 800 e^{0.5057t}
\]
\[
e^{0.5057t} = \frac{30000}{800} = 37.5
\]
\[
0.5057t = \ln(37.5)
\]
\[
t = \frac{\ln(37.5)}{0.5057} \approx 7.504
\]

Thus, the population will reach 30,000 after approximately **7.5 hours**.

---

Would you like more details on any of these steps or have any questions? Here are five related questions:

1. How does changing the initial population affect the growth model?
2. What happens to the growth rate as time increases indefinitely?
3. How would the results change if the growth rate \( k \) were higher or lower?
4. What is the doubling time of the bacterial culture?
5. How would the equation change if there was a limiting population capacity?

**Tip:** In exponential growth, the rate of increase gets faster over time, which can lead to rapid growth after a certain point.

Solve an exponential growth problem where a bacteria culture starts with 800 bacteria and grows to 9000 in 5 hours. Learn to find the population after t hours, calculate the growth rate after 6 hours, and determine when the population will reach 30,000. Ideal for students in Grades 10-12 studying differential equations and exponential growth.

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