can you use normal grade 8,9,10,11 functions rules??? dont use advanced or high level math like you did here. use exponential functions stuff

Sure! Let's simplify the problem and use the basic exponential function rules you might encounter in grades 8-11.

Given the population dataSure! Let's simplify the problem and use the basic exponential function rules you might encounter in grades 8-11.

Given the population 

| Time (weeks) | Population |
|--------------|------------|
| 0            | 600        |
| 1            | 1800       |
| 2            | 5400       |
| 3            | 16200      |

### Part a) Determine an equation to model the data

We notice that the population is growing exponentially. The general form of an exponential growth equation is:

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

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population.
- \( r \) is the growth factor (common ratio).
- \( t \) is the time in weeks.

#### Step 1: Determine \( P_0 \)
From the table, we see that the initial population, \( P_0 \), is 600.

#### Step 2: Determine \( r \)
To find the growth factor \( r \), we can use the population values at successive time intervals. Let's use the population at \( t = 0 \) and \( t = 1 \):

\[ P(1) = P_0 \cdot r \]
\[ 1800 = 600 \cdot r \]
\[ r = \frac{1800}{600} = 3 \]

So the growth factor \( r \) is 3.

Thus, the equation for the population growth model is:
\[ P(t) = 600 \cdot 3^t \]

### Part b) How many ants will be in the colony after 90 days?

First, convert 90 days to weeks:
\[ 90 \text{ days} = \frac{90}{7} \approx 12.86 \text{ weeks} \]

Now, use the model to find \( P(12.86) \):
\[ P(12.86) = 600 \cdot 3^{12.86} \]

Let's calculate this:

Using a calculator:
\[ 3^{12.86} \approx 1424580.497 \]
\[ P(12.86) \approx 600 \cdot 1424580.497 \approx 854748298 \]

Therefore, the population of the ant colony after 90 days will be approximately 854,748,298 ants.

Would you like any more details or explanations?

Here are 5 more questions you might want to ask:
1. How do we identify exponential growth in a data set?
2. What are some examples of exponential growth in real life?
3. How can we estimate the population at a non-integer time point?
4. What happens to the growth factor if the population decreases over time?
5. How can we graph an exponential function?

**Tip:** When working with exponential functions, ensure that you understand the base (growth factor) and how it affects the rate at which the function increases or decreases.

Learn how to model population growth using exponential functions with a detailed example. Explore the basic rules and formulas for exponential functions applied to population data over time. Suitable for students in Grades 8-11.

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