A Population Doubles Every 33 Years. Assuming Exponential Growth Find The Following: (a) The Annual Growth
Understanding population dynamics is crucial in fields ranging from demography and urban planning to environmental science. One common question is how to model population growth when the population doubles over a certain period. In this article, we explore the scenario where a population doubles every 33 years, assuming exponential growth, and focus on calculating the annual growth rate.
Introduction to Exponential Population Growth
Exponential growth describes a process where the quantity increases at a rate proportional to its current value. This model is often used to describe populations under ideal conditions where resources are unlimited, and there are no constraints such as disease, migration, or resource depletion.
Mathematically, the exponential growth model can be expressed as:
\[ P(t) = P_0 \times e^{rt} \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population at time \( t=0 \),
- \( r \) is the growth rate per unit time (e.g., per year),
- \( t \) is the time elapsed.
Given this model, if we know the population doubles over a certain period, we can determine the growth rate \( r \).
Understanding the Doubling Time
The doubling time is the period required for a population to double in size. In our scenario, the doubling time is given as 33 years.
This key piece of information helps us find the growth rate \( r \) by setting \( P(t) = 2 P_0 \) when \( t = 33 \).
Mathematically:
\[ 2 P0 = P0 \times e^{r \times 33} \]
Dividing both sides by \( P_0 \):
\[ 2 = e^{r \times 33} \]
Now, taking the natural logarithm of both sides:
\[ \ln 2 = r \times 33 \]
Therefore, the growth rate \( r \) is:
\[ r = \frac{\ln 2}{33} \]
This formula provides the continuous growth rate per year, which is what we interpret as the "annual growth rate" in the context of exponential growth.
Calculating the Annual Growth Rate
Using the above relation:
\[ r = \frac{\ln 2}{33} \]
Calculate \( \ln 2 \):
\[ \ln 2 \approx 0.6931 \]
Now, divide:
\[ r \approx \frac{0.6931}{33} \approx 0.0210 \]
Expressed as a percentage:
\[ r \approx 2.10\% \]
This means that the population grows by approximately 2.10% each year under the assumption of continuous exponential growth.
Interpreting the Results
The key takeaway is that an exponential growth rate of about 2.10% per year results in the population doubling every 33 years. This insight is useful in various practical contexts:
- Urban Planning: Estimating how quickly a city’s population might increase, aiding infrastructure development.
- Environmental Impact: Understanding how fast populations of certain species or human populations grow, influencing conservation efforts.
- Public Health: Planning for healthcare resources based on expected population growth rates.
Applications and Examples
Let’s explore some practical applications where understanding the annual growth rate is vital.
Example 1: Population Projection
Suppose a town currently has a population of 50,000 residents. If the population doubles every 33 years, what will be the population after 66 years?
Using the exponential growth model:
\[ P(t) = P_0 \times e^{rt} \]
Plugging in the values:
\[ P(66) = 50,000 \times e^{0.0210 \times 66} \]
Calculate the exponent:
\[ 0.0210 \times 66 \approx 1.386 \]
Then:
\[ P(66) = 50,000 \times e^{1.386} \]
Calculate \( e^{1.386} \):
\[ e^{1.386} \approx 4 \]
So,
\[ P(66) \approx 50,000 \times 4 = 200,000 \]
The population would be approximately 200,000 after 66 years, showcasing the power of exponential growth.
Example 2: Estimating Growth Over a Shorter Period
If we want to find the population after 10 years, starting with 100,000 residents:
\[ P(10) = 100,000 \times e^{0.0210 \times 10} \]
Calculate the exponent:
\[ 0.0210 \times 10 = 0.21 \]
Calculate \( e^{0.21} \):
\[ e^{0.21} \approx 1.23 \]
Thus,
\[ P(10) \approx 100,000 \times 1.23 = 123,000 \]
The population would grow to approximately 123,000 in 10 years.
Limitations of the Exponential Model
While exponential growth provides a useful approximation, real-world populations are subject to various limiting factors such as resource availability, environmental constraints, and social changes. As a result, actual growth rates often decrease over time, transitioning into logistic growth models.
Moreover, the assumption of a constant growth rate might not hold in practice due to policy changes, technological advancements, or unforeseen events.
Summary and Conclusion
In this article, we have demonstrated how to determine the annual growth rate of a population that doubles every 33 years under the assumption of exponential growth. The key steps are:
- Express the doubling condition mathematically: \( 2 = e^{r \times 33} \).
- Take the natural logarithm: \( \ln 2 = r \times 33 \).
- Solve for \( r \): \( r = \frac{\ln 2}{33} \approx 0.0210 \) or about 2.10% per year.
This calculation is fundamental in demographic studies, resource planning, and understanding the implications of population growth trends. While the exponential model offers a simplified view, it provides valuable insights into how populations evolve over time under idealized conditions.
Remember: The actual growth rate can vary due to numerous factors, but understanding the basic mathematical relationship helps in making informed projections and analyses.
