Problem 93 - Elementary and Intermediate Algebra

Question

The \(4.3 \%\) annual population growth rate for the Raleigh-Cary metropolitan area in North Carolina is one of the largest of any metropolitan area in the United States. If its growth rate remains constant, how long will it take for its population to double? (Source: U.S. Bureau of the Census)

Step-by-Step Solution

Answer

Approximately 16.28 years.

Step 1: Understanding the problem

We need to find how long it will take for the population to double at a constant annual growth rate. This is an application of exponential growth where we use the rule of 70.

Step 2: Use the Rule of 70

The Rule of 70 is a way to estimate how many years it takes for a quantity to double, given its annual growth rate. The formula is: \( \text{Doubling Time} = \frac{70}{\text{Growth Rate}} \).

Step 3: Substitute the given growth rate into the formula

The given growth rate is 4.3%. Substitute this into the Rule of 70 formula to find the doubling time: \( \text{Doubling Time} = \frac{70}{4.3} \).

Step 4: Calculate the doubling time

Perform the division: \( \frac{70}{4.3} \approx 16.28 \).

Step 5: Interpret the result

This means it will take approximately 16.28 years for the population of the Raleigh-Cary metropolitan area to double, assuming the growth rate remains constant.

Key Concepts

Understanding Population Growth

Population growth is a fundamental concept in demography, which is the statistical study of populations. It refers to the increase in the number of individuals in a population over time. Population growth can be influenced by factors such as birth rates, death rates, and migration patterns.
There are two main types of population growth:

• Exponential growth: This is when the growth rate of the population is proportional to its current size, leading to faster growth over time. It occurs when there are no limiting factors such as food scarcity or space constraints.
• Logistic growth: This is when a population grows rapidly at first, but its growth rate slows as the population reaches its carrying capacity, or the maximum population size that the environment can sustain indefinitely.

In the context of the Raleigh-Cary metropolitan area, exponential growth is being used, meaning the population increases rapidly if the 4.3% growth rate continues without any hindrances.

The Rule of 70 Explained

The Rule of 70 is a quick and easy way to estimate how long it will take for a variable experiencing exponential growth to double. This rule is useful for various fields, including finance and economics, but it is especially useful in understanding population growth.
The formula for the Rule of 70 is:

• Doubling Time = \( \frac{70}{\text{Growth Rate}} \)

Here, the growth rate should be expressed as a percentage.The Rule of 70 simplifies the complexity of exponential calculations and provides an easy-to-use method for predicting growth outcomes. For example, with a constant growth rate of 4.3%, the doubling time for the Raleigh-Cary metropolitan area becomes approximately 16.28 years, according to our calculations.
This rule assumes a constant growth rate, meaning that the conditions for growth do not change significantly over the given period.

Calculating Doubling Time

Doubling time is the period needed for a population to double in size at a constant rate of growth. It is a critical measure in population studies because it reflects how quickly a population is expanding.
To find the doubling time, one can apply the Rule of 70 by dividing 70 by the annual growth percentage rate. For instance, if Raleigh-Cary has an annual growth rate of 4.3%, we find its doubling time by calculating:\[ \text{Doubling Time} = \frac{70}{4.3} \approx 16.28 \text{ years} \]This means that if the population continues to grow by 4.3% per year, it will double in about 16.28 years. Remember, doubling time is significant because it can predict the future size of populations and help with planning resources, infrastructure, and services needed to support such growth.