Problem 97 - Elementary and Intermediate Algebra

Question

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: Understanding the Formula

The formula given is \( t = \frac{\ln 3}{r} \), where \( t \) is the time required for a population to triple, and \( r \) is the continuous growth rate.

Step 2: Converting Percentage to Decimal

Since the growth rate is given as \(12\%\), we need to express it as a decimal. To do this, divide by 100: \( r = \frac{12}{100} = 0.12 \).

Step 3: Plugging Values into the Formula

We substitute \( r = 0.12 \) into the formula: \[ t = \frac{\ln 3}{0.12} \].

Step 4: Calculating Natural Logarithm of 3

Find the natural logarithm of 3, which is approximately \( \ln 3 \approx 1.0986 \).

Step 5: Solving for Time \( t \)

Substitute \( \ln 3 \approx 1.0986 \) into the formula and calculate:\[ t = \frac{1.0986}{0.12} \approx 9.155 \].

Key Concepts

Continuous Growth Rate

When we talk about continuous growth rate, we're referring to a consistent increase in an amount over time. Think about it like compounding interest in a bank account that grows continuously. This is different from simple interest where growth happens in separate steps.

Understanding the formula for continuous growth is essential when analyzing things like population. The formula for the continuous growth rate is closely tied to the formula for exponential growth, which in this case is expressed as:

• \( t = \frac{\ln 3}{r} \)
• \( t \): time it takes for the population to triple
• \( r \): continuous growth rate (as a decimal)

This showcases how continuously growing rates simplify calculating complex exponential processes over time. Importantly, when a population or amount grows continuously at a constant rate, the time it takes to multiply to a certain factor is inversely proportional to the growth rate.

In simple terms, a higher growth rate means the population triples faster, while a lower growth rate means it takes longer.

Natural Logarithm

Natural logarithms help in understanding relationships within exponential growth, particularly with continuous growth rates.

A natural logarithm (\( \ln \)) is a logarithm to the base \( e \), where \( e \) is approximately 2.718 and known as Euler's number. This makes natural logarithms extensively useful in scenarios involving continuous rates of change, like population growth.

To solve problems involving natural logarithms, you need to translate your information into the natural logarithmic form. Take the case from the original exercise. We are asked to determine the time it takes for a town's population to triple.

• Formula: \( t = \frac{\ln 3}{r} \)
• \( \ln 3 \approx 1.0986 \)

Here, \( \ln 3 \) essentially denotes the time factor under a natural logarithmic scale when considering the tripling of the population. Utilizing natural logarithms is crucial because it allows simplification of complex growth equations, making problem-solving in continuous growth scenarios feasible and systematic.

Percentage to Decimal Conversion

When dealing with mathematical formulas that involve percentages, converting them into decimals is crucial. This conversion simplifies calculations and ensures correct application in mathematical operations. In the context of population growth and continuous rates, percentages must be expressed in this form.

Let’s start with the conversion method:

• A percentage is simply a number or ratio expressed as a fraction of 100.
  To convert a percentage to a decimal, you divide the percentage by 100.
• Example: If the growth rate is 12%, the conversion process will be:\( r = \frac{12}{100} = 0.12 \)

Once converted, this decimal form can be readily used in formulas concerned with growth.

In this case, using \( r = 0.12 \) allows seamless substitution into the formula \( t = \frac{\ln 3}{r} \), enabling the calculation of time \( t \) efficiently.

• Why convert? It prevents calculation errors and ensures uniformity when dealing with growth equations.

In nutshell, percentage to decimal conversion is a vital step in mathematical problem-solving, especially in the context of continuous growth equations.