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To solve these questions, we’ll use the exponential growth formula:

\[
N(t) = N_0 e^{kt}
\]

where:
- \( N(t) \) is the population at time \( t \),
- \( N_0 \) is the initial population (the unknown initial size of the culture),
- \( k \) is the growth rate,
- \( t \) is time.

Given dataTo solve these questions, we’ll use the exponential growth formula:

\[
N(t) = N_0 e^{kt}
\]

where:
- \( N(t) \) is the population at time \( t \),
- \( N_0 \) is the initial population (the unknown initial size of the culture),
- \( k \) is the growth rate,
- \( t \) is time.

Given 
1. At \( t = 20 \) minutes, \( N(20) = 300 \).
2. At \( t = 35 \) minutes, \( N(35) = 2000 \).

### Step 1: Solve for \( k \) and \( N_0 \)

Using the formula for exponential growth, set up two equations:

\[
300 = N_0 e^{20k} 
\]
\[
2000 = N_0 e^{35k}
\]

Dividing the second equation by the first will help eliminate \( N_0 \) and solve for \( k \):

\[
\frac{2000}{300} = \frac{N_0 e^{35k}}{N_0 e^{20k}}
\]

Simplify this:

\[
\frac{2000}{300} = e^{15k}
\]

\[
\frac{2000}{300} = \frac{20}{3} \approx 6.6667
\]

Take the natural logarithm on both sides to solve for \( k \):

\[
\ln\left(\frac{20}{3}\right) = 15k
\]

\[
k \approx \frac{\ln\left(\frac{20}{3}\right)}{15}
\]

Calculate this to find \( k \).

### Step 2: Find \( N_0 \)

Once \( k \) is known, substitute back into the equation \( 300 = N_0 e^{20k} \) to solve for \( N_0 \), the initial size of the culture.

### Step 3: Find the Doubling Period

The doubling period \( T \) can be found by solving:

\[
2 = e^{kT}
\]

Taking the natural logarithm on both sides:

\[
\ln(2) = kT
\]

\[
T = \frac{\ln(2)}{k}
\]

### Step 4: Find the Population After 120 Minutes

Use \( N(t) = N_0 e^{kt} \) with \( t = 120 \) minutes to find the population.

### Step 5: Find When the Population Reaches 13,000

Set \( N(t) = 13000 \) and solve for \( t \):

\[
13000 = N_0 e^{kt}
\]

\[
t = \frac{\ln\left(\frac{13000}{N_0}\right)}{k}
\]

Let me know if you'd like further help with calculations, or if you have any questions!

---

Here are 5 follow-up questions to expand your understanding:

1. What is the general formula for calculating exponential growth?
2. How does doubling time relate to the growth constant \( k \)?
3. How would the process differ if the population growth were linear instead of exponential?
4. How does changing \( N_0 \) affect the population growth model?
5. What are some real-life applications of exponential growth models?

**Tip:** Always verify units (like time in minutes or hours) to ensure consistency across calculations.

The count in a bacteria culture was 300 after 20 minutes and 2000 after 35 minutes. Assuming the count grows exponentially:

- What was the initial size of the culture?
- Find the doubling period.
- Find the population after 120 minutes.
- When will the population reach 13000?

Solve an exponential growth problem in a bacteria culture where the count was 300 after 20 minutes and 2000 after 35 minutes. Find the initial population size, doubling period, population after 120 minutes, and time to reach 13000. Suitable for college-level students.

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