Let's solve the problem step by step.

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Part (a): Finding the exponential function

The exponential function has the form:
$f(t) = y_0 \cdot b^t$ where:

• $t$ is the time in years since 2000 ($t = 0$ corresponds to the year 2000),
• $y_0$ is the initial population ($y_0 = 38.4$ million in 2000),
• $b$ is the base, which we will determine using the population data for 2090 ($t = 90$).

The population increases from 38.4 million in 2000 ($t = 0$) to 63.1 million in 2090 ($t = 90$).

1. Substitute into the equation: $f(90) = y_0 \cdot b^{90}$ $63.1 = 38.4 \cdot b^{90}$

2. Solve for $b^{90}$: $b^{90} = \frac{63.1}{38.4} = 1.643229$

3. Solve for $b$ by taking the 90th root (or raising to the power $\frac{1}{90}$): $b = (1.643229)^{\frac{1}{90}}$ Using a calculator: $b \approx 1.0057$

Thus, the exponential function is: $f(t) = 38.4 \cdot (1.0057)^t$

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Part (b): Projected population in 2050 ($t = 50$)

Substitute $t = 50$ into the equation: $f(50) = 38.4 \cdot (1.0057)^{50}$

1. Compute $(1.0057)^{50}$: $(1.0057)^{50} \approx 1.34885$

2. Multiply by 38.4: $f(50) \approx 38.4 \cdot 1.34885 = 51.8$

The projected population in 2050 is: $\boxed{51.8 \, \text{million}}$

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Part (c): Projected population in 2060 ($t = 60$)

Substitute $t = 60$ into the equation: $f(60) = 38.4 \cdot (1.0057)^{60}$

1. Compute $(1.0057)^{60}$: $(1.0057)^{60} \approx 1.42959$

2. Multiply by 38.4: $f(60) \approx 38.4 \cdot 1.42959 = 54.9$

The projected population in 2060 is: $\boxed{54.9 \, \text{million}}$

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Final Results:

1. Exponential Function: $f(t) = 38.4 \cdot (1.0057)^t$

2. Projected Population:

   • 2050: $51.8 \, \text{million}$
   • 2060: $54.9 \, \text{million}$

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Would you like me to explain any step in more detail or perform any additional analysis?

Here are 5 related questions you might consider:

1. How does the population growth rate (represented by $b$) impact long-term predictions?
2. How would doubling or halving the growth rate affect the projections for 2050 and 2060?
3. What happens to the population if the growth is modeled linearly instead of exponentially?
4. How can this model be validated using intermediate census data between 2000 and 2090?
5. How would this population model apply to a scenario with declining growth rates?

Tip: Exponential growth models are sensitive to the base $b$. Small changes in $b$ can lead to large differences in long-term projections. Always interpret predictions with context!