Problem 94 - Intermediate Algebra

Question

The Big Easy. New Orleans has steadily won back some of the population it lost in the wake of Hurricane Katrina in \(2005 .\) If the current \(8.2 \%\) annual increase in population remains constant, how long will it take for its population to double?

Step-by-Step Solution

Answer

It will take approximately 8.54 years for the population to double.

Step 1: Understanding the Problem

We are given that the population of New Orleans is increasing by \(8.2\%\) annually. We need to determine the time it will take for the population to double at this growth rate. This is a classic exponential growth scenario.

Step 2: Using the Doubling Time Formula

The doubling time for exponential growth can be approximated using the rule of 70, which states: \( \text{Doubling Time} = \frac{70}{\text{Growth Rate}} \). The growth rate given here is \(8.2\%\).

Step 3: Applying the Formula

Insert the growth rate into the formula: \( \text{Doubling Time} = \frac{70}{8.2} \). Calculate this to find the time it takes for the population to double.

Step 4: Calculating the Doubling Time

Perform the calculation: \( \frac{70}{8.2} \approx 8.54 \). Thus, it will take approximately 8.54 years for the population to double.

Key Concepts

Doubling Time Formula

In the realm of exponential growth, one of the fundamental tools used is the doubling time formula. This formula gives us an approximate duration for a population or any exponentially growing quantity to double in size. The formula is quite simple and is expressed as:

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

To use it, you insert the annual growth rate into the formula. For instance, if you have a growth rate of 8.2%, you simply divide 70 by 8.2 to get an approximate doubling time. This method is straightforward and is based on the rule of 70, which we'll discuss a bit more in the following sections.

Population Growth

Population growth refers to the increase in the number of individuals in a given area over time. This is often measured annually and can be influenced by factors such as birth rates, death rates, and migration patterns. In the case of New Orleans mentioned in the exercise, the city is experiencing an annual growth rate of 8.2%.
Understanding population growth is crucial for city planning, resource allocation, and economic forecasting. It helps in predicting future demands for housing, infrastructure, and services. In exponential population growth, where the rate of growth is proportional to the size of the population, the doubling time becomes an essential figure for planners and policymakers.

Rule of 70

The rule of 70 is a quick and handy way to calculate the doubling time of an investment or a population growing at a consistent rate. Essentially, this rule states that you can estimate the number of years required to double something by dividing 70 by the annual percentage growth rate.
Here’s a breakdown of why 70 is the chosen number:

• It is derived from the natural logarithm of 2 (about 0.693), which is used in the mathematics of growth rates.
• This natural logarithm approximates to 70 when considering simple percentage-based calculations.

This makes the rule of 70 an accessible option for those who want a quick estimation without deeper mathematical analysis.

Growth Rate Calculation

Calculating the growth rate is a fundamental step in many applications of exponential growth, including our example with the city of New Orleans. The growth rate is typically expressed as a percentage and indicates how quickly a population or investment grows annually.
To find the growth rate:

• Identify the initial and future values of the population or investment.
• Subtract the initial figure from the future figure.
• Divide the result by the initial value.
• Finally, multiply by 100 to convert it into a percentage.

For example, if a population grows from 100 to 108.2 over a year, the growth calculation would be \( \frac{8.2}{100} \times 100 = 8.2\% \). Understanding this calculation is essential for applying the rule of 70 and using the doubling time formula effectively.