Let's solve this step by step, based on the information provided in the image.

(a) What was the deer population in 2003?

The problem states that the population grows exponentially, and that 2003 corresponds to $t = 0$. From the graph, it appears that the initial population in 2003 was 20,000 deer.

(b) Find an exponential growth model $n(t) = C e^{kt}$ for the population.

We are provided with two data points from the graph:

• At $t = 0$ (year 2003), $n(0) = 20,000$.
• At $t = 4$ (year 2007), $n(4) = 37,000$.

The general formula for exponential growth is: $n(t) = C e^{kt}$ At $t = 0$, $n(0) = C$, so we know $C = 20,000$.

Now, using the second data point, $n(4) = 37,000$, we can plug it into the equation to find $k$: $37,000 = 20,000 e^{4k}$ Dividing both sides by 20,000: $\frac{37,000}{20,000} = e^{4k} \quad \Rightarrow \quad 1.85 = e^{4k}$ Taking the natural logarithm of both sides: $\ln(1.85) = 4k \quad \Rightarrow \quad k = \frac{\ln(1.85)}{4}$ Using $\ln(1.85) \approx 0.615$, we find: $k \approx \frac{0.615}{4} = 0.15375$ So the exponential growth model is approximately: $n(t) = 20,000 e^{0.15375 t}$

(c) Use the model to find the percentage rate of change from 2005 to 2006.

To find the percentage rate of change, we need to calculate the population at $t = 2$ (2005) and $t = 3$ (2006), then compute the relative change.

Using the model $n(t) = 20,000 e^{0.15375 t}$:

• For $t = 2$ (2005): $n(2) = 20,000 e^{0.15375 \times 2} \approx 20,000 \times e^{0.3075} \approx 20,000 \times 1.3604 = 27,208$
• For $t = 3$ (2006): $n(3) = 20,000 e^{0.15375 \times 3} \approx 20,000 \times e^{0.46125} \approx 20,000 \times 1.5862 = 31,724$

Now, the percentage rate of change is: $\frac{n(3) - n(2)}{n(2)} \times 100 = \frac{31,724 - 27,208}{27,208} \times 100 \approx \frac{4,516}{27,208} \times 100 \approx 16.6\%$

(b) Growth rate based on the exponential model:

From the exponential model, the growth rate $k$ corresponds to an annual growth of: $k \times 100 = 0.15375 \times 100 \approx 15.38\%$

Compare the two growth rates:

The calculated percentage rate of change from 2005 to 2006 is approximately 16.6%, which is slightly higher than the exponential model's 15.38%.

Would you like further details or clarification? Here are some additional related questions to explore:

1. How would the model change if the growth rate were to decrease over time?
2. What is the doubling time for the deer population based on this exponential model?
3. How could we modify the model if environmental factors limited the population growth?
4. What would be the population in 2010 according to this model?
5. How sensitive is the model to changes in the initial population?

Tip: When calculating exponential growth, always check whether small changes in the growth rate significantly impact future population estimates.