https://file.aiquickdraw.com/mathbot/file/17289687439378rfic1q5.jpg

Let's break down the problem shown in the image, which involves exponential functions related to population growth.

The situation is that the rabbit population in a city park increases by **11% per year**, and the initial population is **283 rabbits**.

### Part (a) - Writing a model for the population \( y \) in terms of years \( t \)
The formula for exponential growth is given by:
\[
y = P(1 + r)^t
\]
Where:
- \( P \) is the initial population (283 rabbits),
- \( r \) is the growth rate (11%, or 0.11),
- \( t \) is the time in years.

So, the model is:
\[
y = 283(1 + 0.11)^t = 283(1.11)^t
\]

### Part (b) - Finding the population in 14 years
To find the population after 14 years, substitute \( t = 14 \) into the formula:
\[
y = 283(1.11)^{14}
\]
Using a calculator:
\[
y \approx 283 \times 3.904 = 1105
\]
The population in 14 years will be approximately **1105 rabbits**.

### Part (c) - Finding the time when the population reaches 11,517
We need to solve for \( t \) when the population \( y = 11517 \):
\[
11517 = 283(1.11)^t
\]
Dividing both sides by 283:
\[
\frac{11517}{283} = (1.11)^t
\]
\[
40.703 = (1.11)^t
\]
Now, take the natural logarithm (ln) of both sides:
\[
\ln(40.703) = t \ln(1.11)
\]
\[
t = \frac{\ln(40.703)}{\ln(1.11)}
\]
Using a calculator:
\[
t \approx \frac{3.707}{0.1044} \approx 35.49
\]
So, it will take approximately **35.490 years** for the rabbit population to reach 11,517.

---

Let me know if you'd like further clarification or more details on any step! Here are 5 related questions to extend your understanding:

1. What would happen to the population model if the growth rate changes to 5% instead of 11%?
2. How does the time required to reach a certain population size change if the initial population was smaller, say 100 rabbits?
3. What is the continuous growth rate formula, and how would it differ from this discrete yearly model?
4. How would the model change if there were factors that decreased the population yearly by a certain percentage?
5. What does the "doubling time" mean in exponential growth, and how could you calculate it for this population?

**Tip:** When solving exponential growth problems, it's important to understand how logarithms are used to solve for time when the equation involves exponentiated terms.

The rabbit population at the city park increases by 11% per year. If there are initially 283 rabbits in the city park:
 a) Write a model for the population (y) in terms of years (t).
 b) Find the rabbit population in 14 years.
 c) How long will it take for the rabbit population to reach 11517?

Learn how to model the exponential growth of a rabbit population in a city park with a yearly increase of 11%. Find the population after 14 years and calculate the time needed for the population to reach 11,517 rabbits. A step-by-step guide with key formulas and exponential functions, suitable for students in Grades 9-11.

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