= 4.2 Correct

The population grows according to an exponential growth model, which can be represented by the formula:

$P(t) = P_0 \cdot r^t$

Where:

• $P(t)$ is the population at time $t$,
• $P_0$ is the initial population,
• $r$ is the common ratio (growth factor),
• $t$ is the time.

Given:

• $P_0 = 3$
• $r = 1.4$

1. Find the explicit formula for $P(t)$:

The general formula for exponential growth is:

$P(t) = 3 \cdot 1.4^t$

So, the explicit formula is:

$P(t) = 3 \cdot 1.4^t$

2. Find the population at $t = 2$:

Now, we can substitute $t = 2$ into the formula to find $P(2)$:

$P(2) = 3 \cdot 1.4^2$ $P(2) = 3 \cdot 1.96 = 5.88$

Thus, the population at $t = 2$ is approximately 5.88.

3. Find the population at $t = 3$:

Now, substitute $t = 3$ into the formula:

$P(3) = 3 \cdot 1.4^3$ $P(3) = 3 \cdot 2.744 = 8.23$

Thus, the population at $t = 3$ is approximately 8.2.

Summary of answers:

• Explicit formula for $P(t)$: $P(t) = 3 \cdot 1.4^t$
• Population at $t = 3$: 8.2

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Would you like more details or have any questions on the steps?

Here are 5 related questions to explore further:

1. How do you determine the growth rate or common ratio from real-world data?
2. How can the exponential growth model be modified to account for limiting factors in a population?
3. What happens to the population if the common ratio $r$ is less than 1?
4. How do you calculate the time it takes for a population to reach a specific size using an exponential model?
5. How does the growth rate affect the long-term behavior of the population?

Tip: To understand the impact of exponential growth better, you can plot the population over time to see how quickly it increases.