To find an exponential model for the population $n(t)$ based on the provided data, we'll proceed as follows:

Step 1: Establish the Model Formula

The exponential model for population decline is given by: $n(t) = n_0 e^{rt}$ where:

• $n_0$ is the initial population (49 million in 2000),
• $r$ is the growth (or decay) rate,
• $t$ is the time in years after 2000.

Step 2: Determine the Decay Rate $r$

We know:

• In 2000 ($t = 0$), $n(0) = 49$ million.
• In 2018 ($t = 18$), $n(18) = 45$ million.

Using the equation: $n(18) = n_0 e^{r \cdot 18}$ we substitute the values to find $r$: $45 = 49 e^{18r}$ Dividing both sides by 49: $\frac{45}{49} = e^{18r}$ Taking the natural logarithm of both sides: $\ln\left(\frac{45}{49}\right) = 18r$ Solving for $r$: $r = \frac{\ln\left(\frac{45}{49}\right)}{18}$ Calculating $r$ to five decimal places:

$r \approx -0.00461$

So the model becomes: $n(t) = 49 e^{-0.00461t}$

Step 3: Estimate the Year When Population Declines to 31 Million

We now use this model to estimate when the population will reach 31 million.

Set $n(t) = 31$ and solve for $t$: $31 = 49 e^{-0.00461t}$ Dividing both sides by 49: $\frac{31}{49} = e^{-0.00461t}$ Taking the natural logarithm of both sides: $\ln\left(\frac{31}{49}\right) = -0.00461t$ Solving for $t$: $t = \frac{\ln\left(\frac{31}{49}\right)}{-0.00461}$ Calculating $t$:

$t \approx 47.99$

This means the population will reach 31 million approximately 48 years after 2000, which is around the year 2048.

Step 4: Find When the Population Will Be Halved

To find when the population will be half of its initial value (i.e., 24.5 million), set $n(t) = 24.5$: $24.5 = 49 e^{-0.00461t}$ Dividing both sides by 49: $\frac{24.5}{49} = e^{-0.00461t}$ Taking the natural logarithm of both sides: $\ln\left(\frac{24.5}{49}\right) = -0.00461t$ Solving for $t$: $t = \frac{\ln\left(\frac{24.5}{49}\right)}{-0.00461}$ Calculating $t$:

$t \approx 150.48$

So the population will be cut in half approximately 150.5 years after the year 2000.

Summary of Answers

1. Exponential model: $n(t) = 49 e^{-0.00461t}$
2. Year when population will decline to 31 million: 2048
3. Years for population to be halved: 150.5 years (around 2150)

Would you like further clarification on any of these calculations?

Here are five related questions to extend your understanding:

1. What assumptions are made when using an exponential decay model for population?
2. How does changing the rate $r$ affect the time when the population halves?
3. What would the model look like if the population were increasing instead?
4. How does this model handle fluctuations in population due to external factors?
5. Can exponential models be applied to other scenarios with a constant percentage change?

Tip: Always check units and initial conditions carefully in growth and decay problems; small errors can lead to significant differences in long-term predictions.