Find an Expression for the number of cells after T hours

To find an expression for the number of cells after \( T \) hours, we can assume that the cells are growing through exponential growth, a common assumption for biological processes like cell division.

Let:

- \( N_0 \) represent the initial number of cells at time \( t = 0 \),
- \( r \) represent the growth rate (in units of per hour),
- \( N(t) \) represent the number of cells after \( t \) hours.

The general formula for exponential growth is:

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

Where:

- \( N(t) \) is the number of cells after \( t \) hours,
- \( N_0 \) is the initial number of cells,
- \( r \) is the growth rate,
- \( e \) is the base of the natural logarithm,
- \( t \) is the time in hours.

### Special Cases:
1. **If cells double every fixed number of hours \( T_d \):**
   The growth rate \( r \) can be expressed as:
   \[
   r = \frac{\ln(2)}{T_d}
   \]
   So the expression becomes:
   \[
   N(t) = N_0 e^{\frac{t \ln(2)}{T_d}}
   \]

2. **If cells divide every hour:**
   The formula simplifies to:
   \[
   N(t) = N_0 \cdot 2^t
   \]

This final formula assumes the cells double in population every hour.

### Do you want more details or have any questions?

Here are 5 relative questions:
1. What is the growth rate \( r \) if the cells double every 2 hours?
2. How does the formula change if the growth is not exponential but linear?
3. How would you estimate \( N_0 \) if you're given the population after \( T \) hours?
4. What happens to the number of cells if the growth rate \( r \) decreases over time?
5. How does this model apply to population growth in other biological contexts?

**Tip:** For exponential growth, knowing the doubling time helps derive the growth rate, a key step in calculating population sizes over time.

Explore how to find an expression for the number of cells after T hours, using exponential growth models. This solution includes formulas for exponential growth, doubling time, and cell division, with key insights into population growth dynamics. Ideal for students in Grades 10-12.

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