Problem 1 - Advanced Engineering Mathematics

Question

The population of a community is known to increase at a rate proportional to the number of people present at time \(t\). If an initial population \(P_{0}\) has doubled in 5 years, how long will it take to triple? To quadruple?

Step-by-Step Solution

Answer

It takes approximately 7.924 years to triple and exactly 10 years to quadruple the population.

Step 1: Understand the Problem

Here we have a population increasing at a rate proportional to its current size. This suggests an exponential growth model. We need to find the time it takes for the population to triple and quadruple given that it doubles in 5 years.

Step 2: Use the Exponential Growth Formula

The formula for exponential growth is given by \( P(t) = P_0 e^{kt} \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( k \) is a constant, and \( e \) is the base of the natural logarithm.

Step 3: Calculate the Constant \( k \)

Given that the population doubles in 5 years: \( 2P_0 = P_0 e^{5k} \).\ Simplifying gives \( 2 = e^{5k} \) or \( \), taking logarithms, \( 5k = \ln(2) \).\ Thus, \( k = \frac{\ln(2)}{5} \approx 0.1386 \).

Step 4: Find the Time for the Population to Triple

For the population to triple: \( 3P_0 = P_0 e^{kt} \).\ Simplifying gives \( 3 = e^{kt} \). Taking the natural log gives \( kt = \ln(3) \).\ Substituting for \( k \), we have \( t = \frac{\ln(3)}{\ln(2)/5} = 5\frac{\ln(3)}{\ln(2)} \approx 7.924 \).

Step 5: Find the Time for the Population to Quadruple

For the population to quadruple: \( 4P_0 = P_0 e^{kt} \).\ This simplifies to \( 4 = e^{kt} \). Solving gives \( kt = \ln(4) \).Substitute \( k \), \( t = \frac{\ln(4)}{\ln(2)/5} = 5\frac{\ln(4)}{\ln(2)} = 10 \).

Key Concepts

Population Growth

Population growth is an essential topic in understanding how communities evolve over time. It refers to the increase in the number of individuals in a population. This growth can be linear or exponential.
Exponential growth implies that the rate of population growth is proportional to the current population size. This means the larger the population, the faster it grows. For example, in the context of our exercise, if the population doubles in 5 years, the time required to further increase varies based on the same growth principle.
In practical scenarios, population growth affects resource management, urban planning, and environmental sustainability.

• Governments need accurate growth predictions for infrastructure planning.
• Underlying factors for growth include fertility rates, mortality rates, and migration patterns.
• Both natural factors (like the availability of resources) and human actions (like policies) influence growth paths.

Logarithms

Logarithms are a potent tool in mathematics, especially when working with exponential growth equations. In simple terms, a logarithm tells you what power you must raise a certain number to get another number.
When you see an equation involving an exponential formula, like population growth, it's often solved using logarithms. For instance, solving for the time when the population triples involves the natural log function, denoted as \( \ln \).
The logarithm simplifies the exponential equation so that we can solve for unknowns easily. This approach allows us to handle exponential relationships effectively:

• Logarithms turn multiplicative processes into additive ones, simplifying calculations.
• A key property is \( \ln(e) = 1 \), making it ideal for working with natural exponential growth.
• Logarithms are crucial for finding constant coefficients and solving for time in growth scenarios.

Exponential Growth Formula

The Exponential Growth Formula is a cornerstone of understanding population dynamics. It mathematically models how populations grow over time, given that growth rate is proportional to the current size.
The standard formula used is \( P(t) = P_0 e^{kt} \), where:

• \( P(t) \) represents the population at time \( t \).
• \( P_0 \) is the initial population size.
• \( e \) is approximately equal to 2.718, the base of natural logarithms.
• \( k \) is the growth constant that determines how fast the population grows.

To find \( k \), you often use known changes over time (like population doubling) to back-calculate using logarithms. Understanding this formula is key as it lays the foundation for predicting future population sizes, analyzing trends, and making informed decisions about growth policy.