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Let’s go through each part of this exponential growth problem step-by-step.

### Given Information:
- Bacteria count is 700 after 10 minutes and 1000 after 35 minutes.
- The growth is exponential, so the general formula for exponential growth is:
  \[
  N(t) = N_0 e^{kt}
  \]
  where:
  - \( N(t) \) is the population at time \( t \),
  - \( N_0 \) is the initial population,
  - \( k \) is the growth rate,
  - \( t \) is time in minutes.

### 1. Initial Size of the Culture
It appears the initial size has already been calculated as approximately \( 606.93 \).

### 2. Finding the Growth Rate \( k \)
Using the data points given (700 at \( t = 10 \) and 1000 at \( t = 35 \)), we can set up two equations to solve for \( k \).

From these, you’ll get a value for \( k \). With \( k \) known, we can proceed to other calculations.

### 3. Doubling Period
The doubling period \( T \) can be found using:
\[
T = \frac{\ln(2)}{k}
\]

### 4. Population after 100 Minutes
Using the exponential formula \( N(100) = N_0 e^{k \cdot 100} \).

### 5. Time to Reach 14000
We can solve for \( t \) when \( N(t) = 14000 \) using:
\[
14000 = 606.93 e^{kt}
\]

Would you like me to go into the calculations for each part in detail? 

---

### Relative Questions
1. How does changing the growth rate \( k \) affect the doubling period?
2. What would happen if the initial size \( N_0 \) was doubled?
3. How does the population growth model change if we assume logistic growth instead of exponential growth?
4. What is the significance of the constant \( e \) in the context of exponential growth?
5. Can you derive an alternative method to calculate the growth rate if given different data points?

### Tip
For exponential growth problems, always identify two time-population points first, as these allow solving for both the initial population and growth rate.

The count in a bacteria culture was 700 after 10 minutes and 1000 after 35 minutes. Assuming the count grows exponentially, find the initial size of the culture, the doubling period, the population after 100 minutes, and when the population will reach 14000.

Solve an exponential growth problem involving bacteria culture where the count reaches 700 in 10 minutes and 1000 in 35 minutes. This problem covers key topics like initial culture size, doubling period, and future population prediction. Suitable for high school students in Grades 10-12.

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