Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

A population is growing at a rate proportional to its size. After 5 years, the population size was 164,000 . After 12 years, the population size was 235,000 . What was the original population size?

Step-by-Step Solution

Answer

The original population size was approximately 120,000.

Step 1: Understand the Problem

We are given that the population grows at a rate proportional to its size, which indicates that the growth follows an exponential model. We also know the population size at 5 years (164,000) and at 12 years (235,000). Our task is to find the initial population size.

Step 2: Write the Exponential Growth Model

The exponential growth model can be expressed as: \[ P(t) = P_0 e^{kt} \]where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( k \) is the growth rate constant, and \( e \) is the base of the natural logarithm.

Step 3: Set Up Equations Using Given Data

We know \( P(5) = 164,000 \) and \( P(12) = 235,000 \). So we can set up two equations:1. \( 164,000 = P_0 e^{5k} \)2. \( 235,000 = P_0 e^{12k} \)

Step 4: Solve for Growth Rate Constant (k)

Divide the second equation by the first equation to eliminate \( P_0 \):\[ \frac{235,000}{164,000} = \frac{P_0 e^{12k}}{P_0 e^{5k}} \]This simplifies to:\[ \frac{235,000}{164,000} = e^{7k} \]Take the natural logarithm of both sides:\[ \ln \left( \frac{235,000}{164,000} \right) = 7k \]Solve for \( k \):\[ k = \frac{1}{7} \ln \left( \frac{235,000}{164,000} \right) \]

Step 5: Substitute k to Find Initial Population (P₀)

Substitute the value of \( k \) back into one of the original equations to find \( P_0 \). Using \( P(5) = 164,000 \):\[ 164,000 = P_0 e^{5k} \]\[ P_0 = \frac{164,000}{e^{5k}} \]Calculate using the value of \( k \) found in the previous step to find \( P_0 \).

Step 6: Calculate Initial Population Size

After computing all values, we find:\( P_0 \approx 120,000 \).This implies the initial population size was approximately 120,000.

Key Concepts

Population Growth

Population growth often follows an exponential model, meaning the rate of growth is proportional to the current population size. You can think of it like interest compounding in a bank account; as the population gets larger, the amount by which it grows increases, too.
In the exercise we're examining, the population growth is described using an exponential formula, which captures how rapidly populations can expand over time. Typically, exponential growth is used in biological contexts, such as in bacteria colony growth, where the resources and space are ample. However, it's important to acknowledge that in real-world scenarios, environmental factors may eventually slow this growth. But in simpler models, such as the given exercise, the exponential model gives us a clear, theoretical way to understand how populations can balloon rapidly under ideal conditions.
Exponential growth can be described mathematically by the formula:

• \[ P(t) = P_0 e^{kt} \]

Here, \( P(t) \) represents the population size at any time \( t \), \( P_0 \) is the initial size, \( k \) is a constant that affects the growth rate, and \( e \) is Euler's number, approximately equal to 2.71828.

Initial Population Size

The initial population size, often denoted as \( P_0 \), is a key value in the exponential growth equation. This is the starting point from which the population grows. In our exercise, we need to determine this initial number to understand how the population expands over time.
Calculating the initial population size involves solving for \( P_0 \) using given data points. By knowing the population at two different times, such as 5 years and 12 years, we can rearrange our exponential formula to isolate \( P_0 \). This is accomplished by substituting known population values into the formula and using algebraic manipulation to solve for the initial size.

• For instance, given \( P(5) = 164,000 \), we use the equation \( 164,000 = P_0 e^{5k} \).

After calculating and substituting the growth rate constant \( k \), we find the initial population \( P_0 \). Solving for \( P_0 \) gives us a reference point from which we can predict future population growth or examine past growth rates. It's like finding the root of a tree to understand how it will grow.

Growth Rate Constant

The growth rate constant, indicated by \( k \) in the exponential growth formula, is critical for understanding how fast a population increases. It's a measure of how quickly or slowly the population size changes over a given period.
To determine \( k \) in our exercise, we use the population data at two different times — specifically at 5 years and 12 years. By solving the equation

• \[ 164,000 = P_0 e^{5k} \]
• \[ 235,000 = P_0 e^{12k} \]

and dividing them to remove \( P_0 \), we isolate a term with \( k \), which allows us to solve for it using logarithms.Using the relation

• \[ rac{235,000}{164,000} = e^{7k} \]

we take the natural log of both sides to solve for \( k \):

• \[ rac{1}{7} ext{ln} igg( rac{235,000}{164,000} igg) = k \]

This gives us a numerical value that reflects how quickly the population grows – a higher \( k \) suggests more rapid growth.
Understanding \( k \) aids scientists and researchers in predicting future populations or comparing different growth scenarios, providing a useful tool for planning and resource allocation.