A population grows exponentially according to the formula P = P₀ × e^(0.05t), where P₀ = 1000 and t is in years. What will the population be after 10 years? (Use e ≈ 2.718) - NBX Soluciones
Exponential Population Growth: How a Population Doubles and Growth Calculations Work
Exponential Population Growth: How a Population Doubles and Growth Calculations Work
Population growth often follows an exponential pattern, especially in ideal conditions with unlimited resources. One of the most widely used mathematical models to describe this is the exponential growth formula:
P = P₀ × e^(rt)
Understanding the Context
Where:
- P is the future population
- P₀ is the initial population
- e is Euler’s number, approximately equal to 2.718
- r is the growth rate per unit time
- t is the time in years
This powerful formula helps demographers, economists, and policymakers predict future population trends based on current data.
Understanding the Formula with Real-World Impact
In the formula P = P₀ × e^(0.05t), the population begins at P₀ = 1000, and grows at a continuous annual rate of 5% (since r = 0.05). Here, t represents time in years. The exponential nature means growth accelerates over time—each year’s increase is based on the new, larger population, not a constant fixed number.
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Key Insights
What Happens After 10 Years?
Let’s compute the population after 10 years using the given values:
- P₀ = 1000
- r = 0.05
- t = 10
- e ≈ 2.718
Start with the formula:
P = 1000 × e^(0.05 × 10)
P = 1000 × e^0.5
Now approximate e^0.5. Since e^0.5 ≈ √2.718 ≈ 1.6487 (using calculator approximation):
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P ≈ 1000 × 1.6487
P ≈ 1648.7
Since population must be a whole number, we round to the nearest whole person:
P ≈ 1649
The Power of Exponential Growth
Over just 10 years, a population of 1,000 grows to approximately 1,649, illustrating the remarkable impact of compound growth. This exponential rise is not linear—it becomes faster as time progresses, making early projections critical for urban planning, healthcare, education, and environmental sustainability.
Summary
Using the exponential growth model P = P₀ × e^(0.05t), with P₀ = 1000 and t = 10:
P ≈ 1,648.7 → rounded to 1,649
Exponential growth reveals how small consistent rates can drive significant long-term change—making accurate population forecasting essential for sustainable development.
Keywords: exponential population growth, exponential modeling, e raised to power, population doubling time, e ≈ 2.718, 10-year population forecast, demographic growth formula
