A research laboratory has been conducting experiments on the rapid increase of cancer cells in an animal. They have observed that cancer cell growth increases by 2% every year with certain medication. Initially, in the year 2018, there were 232.26 units of these cells in the animal.

Using the above data, answer the following questions: Examine the table of values and identify the mathematical function that represents this growth pattern, specifying the key factors of the mathematical function

To determine the mathematical function representing the growth pattern of cancer cells, we start with the given dataTo determine the mathematical function representing the growth pattern of cancer cells, we start with the given 

- Initial number of cancer cells in 2018: \( P_0 = 232.26 \) units
- Annual growth rate: \( 2\% \) (which is \( 0.02 \) as a decimal)

The growth of cancer cells can be modeled by an exponential function because the cells increase by a constant percentage each year. The general form of an exponential growth function is:

\[ P(t) = P_0 \cdot (1 + r)^t \]

Where:
- \( P(t) \) is the number of cancer cells at time \( t \) years after 2018.
- \( P_0 \) is the initial amount of cancer cells (232.26 units).
- \( r \) is the growth rate (0.02).
- \( t \) is the number of years since 2018.

Now, let's write the specific function for this scenario:

\[ P(t) = 232.26 \cdot (1 + 0.02)^t \]

Simplifying further, we get:

\[ P(t) = 232.26 \cdot 1.02^t \]

So, the mathematical function representing the growth pattern of cancer cells is:

\[ P(t) = 232.26 \cdot 1.02^t \]

### Key Factors of the Mathematical Function:
1. **Initial Value (\( P_0 \))**: The initial number of cancer cells, which is 232.26 units.
2. **Growth Rate (\( r \))**: The annual growth rate of the cancer cells, which is 2% or 0.02 in decimal form.
3. **Base of the Exponential Function (\( 1 + r \))**: This is \( 1.02 \), representing the multiplier for the exponential growth each year.
4. **Exponent (\( t \))**: The number of years since 2018, which determines how many times the growth rate is applied.
5. **Function Form**: The exponential function \( P(t) = 232.26 \cdot 1.02^t \), which defines how the number of cancer cells grows over time.

Would you like more details or have any questions about this explanation?

Here are five questions you might consider next:
1. How many cancer cells will there be in 2025?
2. What is the number of cancer cells after 10 years?
3. How long will it take for the number of cancer cells to double?
4. Can you calculate the growth over a specific period, for example, 5 years?
5. How does the growth rate affect the exponential function if it changes?

**Tip**: Always check the units and ensure they are consistent throughout your calculations in mathematical functions.

Explore how cancer cell growth can be modeled using exponential functions with a 2% annual increase. Understand the initial cell count, growth rate, and the function P(t) = 232.26 * 1.02^t. Suitable for advanced high school students studying exponential growth.

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