This exercise uses the exponential growth model.

The population of a country has a relative growth rate of 2% per year. The government is trying to reduce the growth rate to 1%. The population in 2011 was approximately 100 million. Find the projected population (in million people) for the year 2035 for the following conditions. (Round your answers to the nearest million.)

(a)

The relative growth rate remains at 2% per year.

million people

(b)

The relative growth rate is reduced to 1% per year.

million people

The problem you're dealing with involves using the **exponential growth model**, which can be expressed as:

\[
P(t) = P_0 \cdot e^{rt}
\]

where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population,
- \(r\) is the relative growth rate (expressed as a decimal),
- \(t\) is the time in years,
- \(e\) is Euler's number (approximately 2.71828).

We will use this formula to solve both parts of the problem.

---

### (a) **Growth rate remains at 2% per year**:

Given:
- \(P_0 = 100\) million (population in 2011),
- \(r = 0.02\) (2% per year),
- \(t = 2035 - 2011 = 24\) years.

The population in 2035 is:

\[
P(24) = 100 \cdot e^{0.02 \times 24}
\]

\[
P(24) = 100 \cdot e^{0.48}
\]

Using \(e^{0.48} \approx 1.617\):

\[
P(24) \approx 100 \cdot 1.617 = 161.7 \text{ million people}.
\]

Rounding to the nearest million, the population will be approximately **162 million people**.

---

### (b) **Growth rate reduced to 1% per year**:

Given:
- \(P_0 = 100\) million (population in 2011),
- \(r = 0.01\) (1% per year),
- \(t = 24\) years.

The population in 2035 is:

\[
P(24) = 100 \cdot e^{0.01 \times 24}
\]

\[
P(24) = 100 \cdot e^{0.24}
\]

Using \(e^{0.24} \approx 1.271\):

\[
P(24) \approx 100 \cdot 1.271 = 127.1 \text{ million people}.
\]

Rounding to the nearest million, the population will be approximately **127 million people**.

---

### Final Answers:
- (a) The population in 2035 with a 2% growth rate will be approximately **162 million people**.
- (b) The population in 2035 with a reduced 1% growth rate will be approximately **127 million people**.

---

Do you have any questions or would you like a more detailed explanation? Here are 5 related questions to consider:

1. How does the formula for exponential growth change if the growth rate becomes negative (i.e., population decline)?
2. What would happen to the population if the growth rate were reduced further, to 0.5% per year?
3. Can you compare exponential growth to linear growth for the same initial conditions?
4. How would you model a population that reaches a certain maximum limit (logistic growth)?
5. How do external factors, like immigration, affect real-world population growth?

**Tip:** Always ensure the growth rate is expressed as a decimal when plugging it into the exponential growth formula.

The population of a country has a relative growth rate of 2% per year. The government is trying to reduce the growth rate to 1%. The population in 2011 was approximately 100 million. Find the projected population (in million people) for the year 2035 for the following conditions. (Round your answers to the nearest million.) (a) The relative growth rate remains at 2% per year. (b) The relative growth rate is reduced to 1% per year.

Explore how to project the population of a country using the exponential growth model. Given a 2% and 1% growth rate, calculate the population for the year 2035 starting from 100 million in 2011. Suitable for students in Grades 10-12, this problem involves exponential modeling and growth rate analysis.

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