The population of a country dropped from 51.7 million in 1995 to 45.5 million in 2007. Assume  that​ P(t), the​ population, in​ millions, t years after​ 1995, is decreasing according to the exponential decay model.
​a) Find the value of​ k, and write the equation.
​b) Estimate the population of the country in 2017.
​c) After how many years will the population of the country be 3 ​million, according to this​ model?
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Part 1
​a) Select the correct answer below and fill in the answer box to complete your choice.

The population change over time follows the **exponential decay model**, which has the general form:

\[
P(t) = P_0 e^{-kt}
\]

Where:
- \( P(t) \) is the population at time \( t \) (in years),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant (rate of decline),
- \( t \) is the number of years after 1995 (since the population is being measured from 1995).

### **Step 1: Define the variables**
- In 1995, \( P_0 = 51.7 \) million.
- In 2007, \( P(12) = 45.5 \) million (since 2007 is 12 years after 1995).
- We need to find the decay constant \( k \) using the given data.

### **Step 2: Set up the equation**
Using the population at \( t = 12 \):
\[
P(12) = 51.7 e^{-12k}
\]
Substitute \( P(12) = 45.5 \) million:
\[
45.5 = 51.7 e^{-12k}
\]

### **Step 3: Solve for \( k \)**
First, divide both sides of the equation by 51.7 to isolate the exponential term:
\[
\frac{45.5}{51.7} = e^{-12k}
\]

Now take the natural logarithm (ln) of both sides to remove the exponential:
\[
\ln\left(\frac{45.5}{51.7}\right) = -12k
\]

Now solve for \( k \):
\[
k = -\frac{1}{12} \ln\left(\frac{45.5}{51.7}\right)
\]

Let’s compute the value of \( k \).The decay constant \( k \) is approximately \( 0.01065 \).

Thus, the equation for the population model is:

\[
P(t) = 51.7 e^{-0.01065t}
\]

### **Part (b): Estimate the population in 2017**

To estimate the population in 2017, we substitute \( t = 22 \) (since 2017 is 22 years after 1995) into the population model:

\[
P(22) = 51.7 e^{-0.01065 \times 22}
\]

Let's calculate the population for 2017.The estimated population of the country in 2017 is approximately 40.91 million.

### **Part (c): Find when the population will be 3 million**

We now need to determine when the population will reach 3 million. This means solving for \( t \) when \( P(t) = 3 \):

\[
3 = 51.7 e^{-0.01065t}
\]

First, divide both sides by 51.7:

\[
\frac{3}{51.7} = e^{-0.01065t}
\]

Take the natural logarithm of both sides to solve for \( t \):

\[
\ln\left(\frac{3}{51.7}\right) = -0.01065t
\]

Now, solve for \( t \). I'll compute this next.The population will reach 3 million in approximately 267.42 years after 1995, which means around the year **2262**.

### Summary:
- **(a)** The decay constant \( k \) is approximately 0.01065, and the population model is \( P(t) = 51.7 e^{-0.01065t} \).
- **(b)** The estimated population in 2017 is approximately 40.91 million.
- **(c)** The population is expected to reach 3 million about 267.42 years after 1995, which is around the year 2262.

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Do you have any questions or would you like more details on a specific part?

Here are 5 related questions to explore:
1. How does changing the decay constant \( k \) affect the population over time?
2. What would the population be in 2025 according to this model?
3. How can we adjust the model to account for immigration or other external factors?
4. What is the half-life of the population in this decay model?
5. How would the model differ if the population were growing instead of declining?

**Tip:** Exponential models are commonly used to describe both growth and decay processes in fields such as biology, physics, and economics!

The population of a country dropped from 51.7 million in 1995 to 45.5 million in 2007. Assume that​ P(t), the​ population, in​ millions, t years after​ 1995, is decreasing according to the exponential decay model. ​a) Find the value of​ k, and write the equation. ​b) Estimate the population of the country in 2017. ​c) After how many years will the population of the country be 3 ​million, according to this​ model?

Learn how to model a population decline using exponential decay. This problem involves finding the decay constant, estimating future population, and predicting when the population will reach a specific size. The country’s population decreased from 51.7 million in 1995 to 45.5 million in 2007. Detailed step-by-step solution provided.

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