The exponential growth model is given by the formula:

$P(t) = P_0 e^{rt}$

where:

• $P(t)$ is the population at time $t$,
• $P_0$ is the initial population,
• $r$ is the growth rate,
• $t$ is the time in years (or other appropriate time unit),
• $e$ is the base of the natural logarithm (approximately 2.71828).

In this case, we are given:

• $P_0 = 15$,
• $r = 0.15$,
• $t = 1$ (since we are asked to find $P_1$, the population after 1 time unit).

Now, substitute the values into the formula:

$P_1 = 15 e^{0.15 \times 1}$

First, calculate the exponent:

$P_1 = 15 e^{0.15}$

Using $e^{0.15} \approx 1.1618$:

$P_1 = 15 \times 1.1618 \approx 17.427$

So, the population after 1 time unit is approximately:

$P_1 \approx 17.43$

Would you like more details or further clarification?

Here are some related questions you might find interesting:

1. How would the population change after 5 years, $P_5$?
2. How do different growth rates impact the population over time?
3. What happens if the growth rate is negative (i.e., decay)?
4. How does the formula change for a population that grows according to a logistic model?
5. How do you interpret the value of $r$ in the exponential growth model?

Tip: When working with exponential growth or decay models, it's helpful to graph the function to visualize how quickly the population increases (or decreases) over time.