The problem gives an exponential decay model for the population, which follows the general form of the equation:

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

Where:

• $P(t)$ is the population at time $t$ (in years after 1995),
• $P_0$ is the initial population (in 1995),
• $k$ is the decay constant (to be found),
• $t$ is the number of years after 1995.

Part (a) Find the value of $k$

We are given:

• $P_0 = 51.6516$ million (population in 1995),
• $P(t) = 44.5445$ million (population in 2008, which is 13 years after 1995, so $t = 13$).

We substitute the known values into the equation:

$44.5445 = 51.6516 e^{13k}$

Now solve for $k$.

1. Divide both sides by 51.6516:

$\frac{44.5445}{51.6516} = e^{13k}$

2. Take the natural logarithm of both sides:

$\ln\left(\frac{44.5445}{51.6516}\right) = 13k$

3. Solve for $k$:

$k = \frac{1}{13} \ln\left(\frac{44.5445}{51.6516}\right)$

Let's calculate $k$.

I'll compute that now.The decay constant $k$ is approximately $-0.0114$ (rounded to four decimal places).

Thus, the equation for the population model is:

$P(t) = 51.6516 e^{-0.0114t}$

Part (b) Estimate the population in 2020

For 2020, the time elapsed since 1995 is $t = 2020 - 1995 = 25$ years. We will now estimate the population by plugging $t = 25$ into the equation:

$P(25) = 51.6516 e^{-0.0114 \times 25}$

Let me calculate that next.The estimated population of the country in 2020 is approximately $38.86$ million.

Part (c) When will the population be 33 million?

We are given that $P(t) = 33$ million, and we need to find $t$. Using the equation:

$33 = 51.6516 e^{-0.0114t}$

1. Divide both sides by 51.6516:

$\frac{33}{51.6516} = e^{-0.0114t}$

2. Take the natural logarithm of both sides:

$\ln\left(\frac{33}{51.6516}\right) = -0.0114t$

3. Solve for $t$:

$t = \frac{\ln\left(\frac{33}{51.6516}\right)}{-0.0114}$

I'll calculate that now.The population will reach 33 million approximately 39.34 years after 1995, which means around the year 2034.

Summary:

• (a) The decay constant $k$ is $-0.0114$, and the population model is $P(t) = 51.6516 e^{-0.0114t}$.
• (b) The estimated population in 2020 is approximately 38.86 million.
• (c) The population will reach 33 million around the year 2034.

Would you like more details on any part of the solution?

Here are some related questions for further exploration:

1. How does the exponential decay model compare to linear population decline models?
2. How does changing the value of $k$ affect the population estimate in 2020?
3. If the decay constant $k$ were smaller, how would it impact the time it takes for the population to reach 33 million?
4. What would happen if population growth were modeled instead of decay?
5. Can this model accurately predict population far into the future?

Tip: Exponential models are powerful for estimating growth or decay, but real-world factors can sometimes deviate from these predictions over time.