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.