Problem 41 - Intermediate Algebra

Question

Suppose that the equation \(P(t)=P_{0} e^{0.02 t}\), where \(P_{0}\) represents an initial population, and \(t\) is the time in years, is used to predict population growth. How long does this equation predict it would take a city of 50,000 to double its population?

Step-by-Step Solution

Answer

It takes approximately 34.66 years to double the population.

Step 1: Understand the Problem

We need to determine the time \(t\) it takes for the population to double. The initial population \(P_0\) is 50,000 according to the problem and we want to find \(t\) such that the population \(P(t)\) is 100,000.

Step 2: Set Up the Doubling Equation

The population needs to double, so set \(P(t) = 2P_0\). Plug this into the given population growth equation:\[2P_0 = P_0 e^{0.02t}\].

Step 3: Simplify the Equation

Divide both sides by \(P_0\) to simplify the equation:\[2 = e^{0.02t}\].

Step 4: Solve for \(t\)

To isolate \(t\), take the natural logarithm (ln) of both sides:\[\ln(2) = 0.02t\].Next, solve for \(t\) by dividing both sides by 0.02:\[t = \frac{\ln(2)}{0.02}\].

Step 5: Calculate the Time

Now compute the value of \(t\):\[t = \frac{\ln(2)}{0.02} \approx 34.657\].Since \(t\) is a time period in years, it will take approximately 34.66 years for the population to double.

Key Concepts

Population Doubling

Population doubling is a key concept in understanding exponential growth patterns. When a population size increases by 100%, it is said to have doubled. This is particularly useful when studying populations, be it in ecology, for cities, or microorganisms. For example, if a city's population grows from 50,000 to 100,000, it has doubled.

In the context of our problem, population doubling can be observed through the growth equation where we want the final population to be twice the initial one. This emphasis on doubling is relevant for planners and scientists as it helps in making predictions and planning resources accordingly.

Understanding the time required for doubling also helps in gauging the sustainability of growth over time.

Natural Logarithm

The natural logarithm, often denoted as ln, is a way to express powers of the constant e, approximately equal to 2.71828. It is useful because it helps us solve equations involving exponential growth, making complex equations manageable.

When dealing with population doubling using the equation \(2 = e^{0.02t}\), taking the natural logarithm of both sides lets us solve for \(t\). This step simplifies the exponential function since \(\ln(e^x) = x\). Hence, we convert \(2 = e^{0.02t}\) into \(\ln(2) = 0.02t\), allowing us to isolate \(t\) and solve the equation.

This conversion from exponential form to logarithmic form is a vital tool for breaking down exponential equations into solvable steps.

Time Calculation

Time calculation in exponential growth scenarios helps us figure out how long certain processes will take. For our population example, once we have the equation \(\ln(2) = 0.02t\), we can calculate the time \(t\).

We isolate \(t\) by dividing both sides of the equation by 0.02, getting \(t = \frac{\ln(2)}{0.02}\). This computation gives us the exact time it will take for the population to double. By completing this calculation, we find that it takes approximately 34.66 years for the city's population to reach 100,000 from 50,000.

This type of calculation is critical in growth projections and allows for precise planning and forecasting.

Growth Equation

The growth equation is fundamental to understanding how populations increase over time. Our specific equation \(P(t)=P_{0} e^{0.02 t}\) represents population growth based on an initial population \(P_0\) and a continuous growth rate expressed as a power of \(e\).

In this formula, \(P_0\) represents the starting population, \(e\) is the base of the natural logarithm, and \(0.02t\) indicates the growth rate and time. This represents continuous exponential growth, where the population size multiplies by itself at a steady rate over time.

The elegance of this equation is in its simplicity and applicability to real-world scenarios, making it a go-to model for predicting how populations change and grow under consistent growth rates.