Problem 97 - Intermediate Algebra

Question

Population Growth. \(\quad\) A population growing continuously at an annual rate \(r\) will triple in a time \(t\) given by the formula \(t=\frac{\ln 3}{r} .\) How long will it take the population of a town to triple if it is growing at the rate of \(12 \%\) per year?

Step-by-Step Solution

Answer

It will take approximately 9.16 years for the population to triple.

Step 1: Identify Given Values

We are given that the continuous growth rate \( r \) is \( 12\% \), which is equivalent to \( r = 0.12 \) when expressed as a decimal, and we want to find the time \( t \) required for the population to triple.

Step 2: Use the Provided Formula

The formula provided for the time \( t \) it takes for a population to triple is \( t = \frac{\ln 3}{r} \). We will use this formula with the given growth rate \( r = 0.12 \).

Step 3: Calculate the Natural Logarithm

Calculate \( \ln 3 \), which is approximately 1.0986. This value represents the natural logarithm of 3, and will be used in our formula for \( t \).

Step 4: Plug in the Values

Substitute \( r = 0.12 \) and \( \ln 3 \approx 1.0986 \) into the formula: \[ t = \frac{1.0986}{0.12} \].

Step 5: Perform the Division

Calculate \( t = \frac{1.0986}{0.12} \) which equals approximately 9.155. This calculation gives us the time in years for the population to triple.

Key Concepts

Population Growth

Population growth is a fascinating concept that relates to how the number of organisms in a population increases over time. In the context of our exercise, we're exploring how a town's population grows when it expands continuously. Population growth can occur at different rates. In cases of exponential growth, the population grows proportionally to the current population size. When growth is continuous, it means there is no stop-and-go; the population is always increasing at every moment.
To understand this idea, imagine a small town. If this town experiences continuous exponential growth, its population won't just grow at specific intervals, like year-end or quarterly. Instead, the growth happens without interruption, which is precisely what our formula for tripling the population helps represent.

Natural Logarithms

Natural logarithms, often denoted as \( \ln \), are a specific type of logarithm. Instead of being based on the number 10 like common logarithms, natural logarithms use the base \(e\), where \(e\) is roughly equal to 2.71828. The natural logarithm is particularly useful in situations involving continuous growth, such as population growth.
In our problem, the natural logarithm of 3 is used—represented as \( \ln 3 \). This value (approximately 1.0986) plays a key role because it helps determine how long it takes for a population to triple. It's crucial to understand natural logarithms because they simplify complex exponential equations into linear forms, making calculations like these more manageable.

Continuous Growth Rate

When we talk about a continuous growth rate, we describe a scenario where the rate of increase is steady—unhindered by external disruptions—and occurs at every possible moment. This kind of growth is calculated as a percentage and is converted to a decimal for mathematics applications.
In practical terms, a continuous growth rate allows for more accurate predictions over time, as it considers every second of growth rather than specific intervals. For our town's population, the given continuous growth rate of 12% is expressed as 0.12 in calculations. Using this constant rate in formulas, we can predict how long it will take for a population to grow by a certain amount, like tripling, by employing the formula \( t = \frac{\ln 3}{r} \). This formula empowers us to precisely understand the length of time a steady, unchanging growth rate will affect population numbers.