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The question provides a population model:

\[
N(t) = 8.3 \cdot (1.33)^t
\]

where \( N \) represents the population in millions, and \( t \) is measured in decades starting from \( t = 0 \), which corresponds to the year 1815.

To solve this question, we need to:

1. **Calculate the population growth rate in 1815 (at \( t = 0 \))**.
2. **Calculate the population growth rate in 1865 (at \( t = 5 \), five decades later)**.

### Step 1: Calculate \( N(t) \) and its Growth Rate
To find the growth rate of the population at any given time, we’ll differentiate \( N(t) \) with respect to \( t \).

Given:
\[
N(t) = 8.3 \cdot (1.33)^t
\]

The derivative \( N'(t) \) gives the rate of growth at time \( t \):

\[
N'(t) = 8.3 \cdot (1.33)^t \cdot \ln(1.33)
\]

#### Substitute \( t = 0 \) (Year 1815)
For \( t = 0 \):
\[
N(0) = 8.3 \cdot (1.33)^0 = 8.3 \text{ million}
\]
\[
N'(0) = 8.3 \cdot (1.33)^0 \cdot \ln(1.33) = 8.3 \cdot \ln(1.33)
\]

#### Substitute \( t = 5 \) (Year 1865)
For \( t = 5 \):
\[
N(5) = 8.3 \cdot (1.33)^5
\]
\[
N'(5) = 8.3 \cdot (1.33)^5 \cdot \ln(1.33)
\]

### Step 2: Compute and Compare
Now, let’s compute the values using \(\ln(1.33) \approx 0.285\).

1. **Population growth rate in 1815 (at \( t = 0 \))**:
   \[
   N'(0) = 8.3 \cdot 0.285 \approx 2.3655 \text{ million per decade}
   \]

2. **Population growth rate in 1865 (at \( t = 5 \))**:
   \[
   N(5) \approx 8.3 \cdot (1.33)^5 \approx 8.3 \cdot 4.268 = 35.4244 \text{ million}
   \]
   \[
   N'(5) \approx 35.4244 \cdot 0.285 \approx 10.0909 \text{ million per decade}
   \]

### Conclusion
The population growth rate increased from approximately \( 2.37 \) million per decade in 1815 to about \( 10.09 \) million per decade in 1865, reflecting a substantial increase over these five decades.

Would you like more detailed explanations on any part of this solution?

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Here are 5 questions that build on this concept:

1. What is the population in 1875 (six decades from 1815) according to this model?
2. How would the population growth rate change if the growth factor were different?
3. How does exponential growth affect population size compared to linear growth?
4. Can you derive the doubling time for this population model?
5. How might we adjust this model to reflect a slowing growth rate over time?

**Tip:** Exponential growth models are common in population studies, finance, and other fields. The base of the exponential function indicates the growth factor per time period.

At the beginning of Section 1.4, we saw that the population model N(t) = 8.3(1.33)^t, where N is in millions and t is in decades starting at t = 0 representing the year 1815, can be used to describe the size of the U.S. population during most of the nineteenth century. Use this relationship to compare the total rate of growth of the U.S. population in 1815 versus the growth rate in 1865 (i.e., five decades later).

Learn how to use an exponential growth model to compare the U.S. population growth rates in 1815 and 1865. This problem explores population modeling, differentiation, and exponential growth concepts, suitable for students in Grades 11-12 or college level.

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