Problem 94 - Elementary and Intermediate Algebra

Question

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? (Source: money.cnn.com)

Step-by-Step Solution

Answer

It will take about 8.8 years for the population to double at an annual growth rate of 8.2%.

Step 1: Understand the Problem

We need to find out how long it will take for a population to double if it is increasing at a constant rate of \(8.2\%\) per year.

Step 2: Use the Population Doubling Formula

The formula to calculate the time \(t\) it takes for a population to double at a constant percentage increase per year is: \[ t = \frac{\ln(2)}{\ln(1 + r)} \]where \(r\) is the growth rate expressed as a decimal. Here, \(r = 0.082\).

Step 3: Substitute and Solve for Time

Substitute \(r = 0.082\) into the formula:\[ t = \frac{\ln(2)}{\ln(1.082)} \]Calculate the natural logarithms:\[ \ln(2) \approx 0.6931 \quad \text{and} \quad \ln(1.082) \approx 0.0788 \]Now solve for \(t\):\[ t = \frac{0.6931}{0.0788} \approx 8.8 \]

Step 4: Interpret the Result

The calculation shows it will take approximately \(8.8\) years for the population to double if the annual growth rate remains constant at \(8.2\%\).

Key Concepts

Exponential Growth

Exponential growth describes a situation where a quantity increases at a consistent rate over time. This type of growth does not just mean getting bigger; it means getting bigger at an accelerating rate. This concept is crucial in understanding population changes, among other things.
In the case of New Orleans' population, the exponential growth occurs because a steady percentage (8.2% in this instance) is being added to the population each year. After the first year, the population grows by 8.2% of the starting number. By the next year, it grows again by 8.2%, but this time on the new total, which includes the previous growth. This compounding effect results in an increasingly rapid growth rate over time.
One way to easily remember exponential growth is by thinking about the idea that growth compounds or "expands" exponentially, rather than linearly. This means the increase is not just faster but gets even faster as time goes on, especially noticeable in long-term situations. This characteristic makes exponential growth a key concept in various fields, including finance, biology, and of course, population studies.

Natural Logarithm

To work with exponential growth, we often use logarithms, specifically natural logarithms. A natural logarithm, denoted as \( \ln \), is a mathematical function that helps us reverse the process of exponentiation. Simply put, it tells us how many times we must multiply a number by itself to reach a certain value.
In the context of population growth, natural logarithms enable us to solve the doubling time formula. For instance, \( \ln(2) \) is critical when calculating how long it takes for a population to double because it represents the point at which growth has doubled.
Computing natural logarithms can seem intricate, but they are quite simple with modern calculators. Just as multiplication is the opposite of division, natural logarithms work hand-in-hand with exponential growth. They allow us to solve for time, such as determining how many years it will take at a certain growth rate until a population reaches twice its size.

Growth Rate Calculation

In order to solve population growth problems, it’s essential to correctly understand and calculate the growth rate. The growth rate is often given as a percentage and needs to be converted to a decimal to work in mathematical formulas.
For a given percentage, converting it to a decimal involves dividing by 100. For example, an 8.2% growth rate translates to 0.082 when converted. This decimal is what we refer to as \( r \) in growth formulas, such as in the population doubling formula \( t = \frac{\ln(2)}{\ln(1 + r)} \). Hence, precision in this conversion step is crucial for accurate calculations.
Once you have the correct decimal representation of the growth rate, it becomes straightforward to use it in equations that predict how quickly something grows or doubles. Growth rates are not only applicable in population studies but also have significant roles in areas like economics and finance, where they might forecast company growth, investment returns, or inflation impacts over time.