To model the growth of the elk herd using the discrete growth model, we use the given formula:

$A(t) = I(1 + r)^t$

where:

• $A(t)$ is the population after $t$ years.
• $I$ is the initial population.
• $r$ is the growth rate (as a decimal).
• $t$ is the number of years.

Given:

• $I = 112$ (the initial population of elk)
• $r = 0.08$ (the growth rate of 8% per year)
• $t$ is the number of years.

We substitute these values into the formula:

$A(t) = 112(1 + 0.08)^t$

$A(t) = 112(1.08)^t$

To find the population of the elk herd after a specific number of years, we substitute $t$ with the desired number of years. For example, to find the population after 5 years:

$A(5) = 112(1.08)^5$

Calculate $(1.08)^5$:

$(1.08)^5 \approx 1.46933$

Now, multiply by the initial population:

$A(5) = 112 \times 1.46933$

$A(5) \approx 164.56$

Therefore, the population of the elk herd after 5 years is approximately 165 elk (since we typically round to the nearest whole number for population counts).

Would you like more details or have any questions?

Here are 5 more questions you can ask:

1. How does the discrete model differ from the continuous growth model in population studies?
2. What would be the population of the elk herd after 10 years?
3. How do different growth rates affect the population size over time?
4. Can you explain the derivation of the exponential growth formula?
5. What factors could affect the accuracy of this population model in a real-world scenario?

Tip: When using exponential growth models, it's important to consider the impact of growth rates and initial values on the long-term behavior of the population. Small changes in these parameters can lead to significantly different outcomes.