We are given an exponential decay model for the population $P(t)$, where $t$ is the number of years after 1995. The population decreases from 52.5 million in 1995 to 44.1 million in 2008. We need to determine:

Part a) Find the value of $k$ and write the equation.

The general form of an exponential decay model is:

$P(t) = P_0 e^{-kt}$

Where:

• $P_0$ is the initial population at $t = 0$ (which corresponds to 1995),
• $k$ is the decay constant, and
• $t$ is the time in years after 1995.

From the problem, we know:

• $P_0 = 52.5$ million (population in 1995),
• $P(13) = 44.1$ million (population in 2008, which is 13 years after 1995).

We can use this information to solve for $k$. Substituting into the exponential model:

$44.1 = 52.5 e^{-13k}$

Now, solve for $k$:

1. Divide both sides by 52.5:

$\frac{44.1}{52.5} = e^{-13k}$

$0.84 = e^{-13k}$

2. Take the natural logarithm (ln) of both sides:

$\ln(0.84) = -13k$

3. Solve for $k$:

$k = -\frac{\ln(0.84)}{13}$

Let's compute this:

$k = -\frac{\ln(0.84)}{13} \approx -\frac{-0.1744}{13} \approx 0.0134$

Thus, $k \approx 0.0134$.

So, the population model is:

$P(t) = 52.5 e^{-0.0134t}$

Part b) Estimate the population in 2019.

2019 is 24 years after 1995, so $t = 24$.

Substitute $t = 24$ into the population model:

$P(24) = 52.5 e^{-0.0134 \times 24}$

$P(24) = 52.5 e^{-0.3216}$

Now, compute $P(24)$:

$P(24) = 52.5 \times e^{-0.3216} \approx 52.5 \times 0.7251 \approx 38.0$

So, the estimated population of the country in 2019 is approximately 38.0 million.

Part c) After how many years will the population be 3 million?

We are looking for $t$ when $P(t) = 3$. Using the equation:

$3 = 52.5 e^{-0.0134t}$

Solve for $t$:

1. Divide both sides by 52.5:

$\frac{3}{52.5} = e^{-0.0134t}$

$0.0571 = e^{-0.0134t}$

2. Take the natural logarithm of both sides:

$\ln(0.0571) = -0.0134t$

$-2.862 = -0.0134t$

3. Solve for $t$:

$t = \frac{-2.862}{-0.0134} \approx 213.7$

So, it will take approximately 214 years for the population to reach 3 million, which would occur in the year:

$1995 + 214 = 2209$

Thus, the population will be 3 million around the year 2209.

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Summary of answers:

a) The value of $k$ is approximately 0.0134, and the equation is:

$P(t) = 52.5 e^{-0.0134t}$

b) The population in 2019 is approximately 38.0 million.

c) The population will reach 3 million around the year 2209, which is approximately 214 years after 1995.

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Let me know if you'd like more details or have further questions!

Follow-up questions:

1. How would the equation change if the population growth was instead modeled as exponential growth?
2. Can you explain the difference between exponential growth and exponential decay?
3. What are some real-world examples of exponential decay outside of population models?
4. How does the decay constant $k$ affect the rate of population change?
5. What would happen if the population started at a different value, say 60 million in 1995?

Tip: For solving exponential equations like these, always start by isolating the exponential term before applying logarithms.