The number of cells in a tumor doubles every 4.5 months. If the tumor begins with a single cells, how many cells will there be after 4 years? After 6 years?

The growth of the number of cells in the tumor follows an exponential model, since the number of cells doubles at regular intervals. The general formula for exponential growth is:

\[
N(t) = N_0 \cdot 2^{t/T}
\]

Where:
- \(N(t)\) is the number of cells at time \(t\),
- \(N_0\) is the initial number of cells,
- \(T\) is the doubling time (in months), and
- \(t\) is the time that has passed (in months).

### Given:
- Initial number of cells, \(N_0 = 1\),
- Doubling time, \(T = 4.5\) months,
- We want to find \(N(t)\) after 4 years and after 6 years.

First, convert years to months:
- 4 years = \(4 \times 12 = 48\) months,
- 6 years = \(6 \times 12 = 72\) months.

Now, apply the formula:

1. **After 4 years (48 months):**

\[
N(48) = 1 \cdot 2^{48/4.5}
\]
\[
N(48) = 2^{10.67} \approx 1623.5
\]

So, after 4 years, there will be approximately 1,624 cells.

2. **After 6 years (72 months):**

\[
N(72) = 1 \cdot 2^{72/4.5}
\]
\[
N(72) = 2^{16} = 65536
\]

After 6 years, there will be 65,536 cells.

### Final answers:
- After 4 years, approximately **1,624 cells**.
- After 6 years, exactly **65,536 cells**.

Do you want more details or have any questions?

---

1. How does exponential growth differ from linear growth?
2. Can we estimate the number of cells after 5 years using the same method?
3. What would happen if the doubling time changed?
4. How can we find the time at which the number of cells reaches a certain threshold?
5. How does the doubling time affect the growth rate?

**Tip:** Exponential growth results in much faster increases over time than linear growth, especially with smaller doubling periods.

The number of cells in a tumor doubles every 4.5 months. If the tumor begins with a single cell, how many cells will there be after 4 years? After 6 years?

This problem explores exponential growth in the context of tumor cells doubling every 4.5 months. Learn how to calculate the number of cells after 4 and 6 years using the formula N(t) = N_0 * 2^(t/T). The solution explains key concepts in exponential growth and is suitable for students in Grades 9-12.

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