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.

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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.

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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.

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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.

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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.