Consider two populations, A and B. Population A begins with a size of 2040, while population B begins with a size of 120.

Population A increases by 15 each year, while population B grows by 15% each year.

Which population will be larger after 20 years? Explain.
(You must show some details and justify your answer to receive full credit!)

Let's analyze the growth of both populations step by step over 20 years.

### Given:
- **Population A**: 
  - Initial size: \( P_A(0) = 2040 \)
  - Increases by **15** each year (constant growth).

- **Population B**: 
  - Initial size: \( P_B(0) = 120 \)
  - Grows by **15%** each year (exponential growth).

We want to determine which population will be larger after 20 years.

---

### Step 1: Growth of Population A

Population A grows linearly by 15 each year, so we can model its growth using the formula for **linear growth**:
\[
P_A(n) = P_A(0) + r_A \cdot n
\]
Where:
- \( P_A(0) = 2040 \) is the initial size,
- \( r_A = 15 \) is the constant increase per year,
- \( n \) is the number of years.

After 20 years (\( n = 20 \)):
\[
P_A(20) = 2040 + 15 \times 20 = 2040 + 300 = 2340
\]
So, after 20 years, **Population A will have 2340 people**.

---

### Step 2: Growth of Population B

Population B grows exponentially by 15% each year. The formula for **exponential growth** is:
\[
P_B(n) = P_B(0) \times (1 + r_B)^n
\]
Where:
- \( P_B(0) = 120 \) is the initial size,
- \( r_B = 0.15 \) is the growth rate (15% per year),
- \( n \) is the number of years.

After 20 years (\( n = 20 \)):
\[
P_B(20) = 120 \times (1.15)^{20}
\]
Now, let's calculate \( (1.15)^{20} \), which is approximately **16.366**.

So:
\[
P_B(20) = 120 \times 16.366 \approx 1963.92
\]
Rounding to the nearest whole number, **Population B will have approximately 1964 people** after 20 years.

---

### Step 3: Comparison of Populations After 20 Years

- **Population A after 20 years**: 2340 people.
- **Population B after 20 years**: 1964 people.

Thus, **Population A will be larger than Population B** after 20 years.

---

### Conclusion:

After 20 years, Population A will have **2,340 people**, while Population B will have **1,964 people**. Therefore, **Population A will still be larger** than Population B after 20 years, despite Population B growing at a faster exponential rate.

Would you like further details or any clarifications?

### Related questions:
1. How many years will it take for Population B to exceed Population A?
2. What would happen if Population A increased by 25 each year instead of 15?
3. What is the population of Population B after 10 years?
4. How many people would Population A have after 30 years?
5. What if Population B grew by 20% per year instead of 15%?

### Tip:
Linear growth adds the same amount every year, while exponential growth multiplies by a percentage, so exponential growth eventually overtakes linear growth for long periods of time.

Consider two populations, A and B. Population A begins with a size of 2040, while population B begins with a size of 120. Population A increases by 15 each year, while population B grows by 15% each year. Which population will be larger after 20 years? Explain.

Analyze the growth of two populations, A and B, where A grows linearly by 15 each year and B grows exponentially by 15% each year. Understand which population will be larger after 20 years using step-by-step calculations. Suitable for Grades 10-12.

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