Problem 22 - Calculus for Biology and Medicine

Question

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population whose size increases by \(50 \%\) in each unit of time and that has 32 individuals at time 0 .

Step-by-Step Solution

Answer

The population size as a function of time is \( N_t = 32 \times e^{0.5t} \).

Step 1: Understanding the Exponential Growth Model

Exponential growth is characterized by the formula \( N_t = N_0 \times e^{rt} \), where \( N_t \) is the population at time \( t \), \( N_0 \) is the initial population size, and \( r \) is the growth rate. Our goal is to find \( N_t \) as a function of \( t \).

Step 2: Finding the Growth Rate

The population increases by 50% in each unit of time. We express the growth rate in decimal form: \( 50\% = 0.5 \). Hence, \( r = 0.5 \).

Step 3: Determine Initial Conditions

The initial population size \( N_0 \) is given as 32 individuals. Substitute \( N_0 = 32 \) into the exponential growth formula.

Step 4: Formulating the Exponential Growth Equation

Substitute \( N_0 = 32 \) and \( r = 0.5 \) into the exponential growth formula: \[ N_t = 32 \times e^{0.5t} \]. Thus, this is the equation representing the population size as a function of time.

Key Concepts

Population Size

Understanding population size is fundamental when dealing with concepts of growth, especially in scenarios like exponential growth. Population size refers to the total number of individuals within a defined area at a particular time. In our example, this is denoted by \(N_t\), which is the population size at a given time \(t\). It is crucial to determine the starting point of the population, known as the initial population size or \(N_0\). In many real-world situations, understanding the scale of the initial population is vital for predicting future growth. In our exercise, \(N_0\) was given as 32 individuals. Hence, at time zero (the beginning of our observation), our population size is 32 individuals. By knowing \(N_0\), we can apply this to the exponential growth formula to project the population size at future times.

Growth Rate

The growth rate is a crucial factor in determining how a population increases over time. It represents the rate at which the population size changes over a specific period, expressed as a proportion of the initial population. In the context of exponential growth, the growth rate \(r\) is the constant that determines the speed of population increase.For our given scenario, the growth rate was described as an increase of 50% per unit of time. To use this in mathematical formulas, we convert the percentage into a decimal, resulting in a growth rate of 0.5. This value indicates that with each unit of time, the population grows by half its previous size. Accurately determining \(r\) is essential for crafting the exponential growth equation and predicting future population sizes.

Exponential Growth Formula

The exponential growth formula is a mathematical representation used to calculate how a population grows over time, assuming a constant growth rate. This formula is represented by \(N_t = N_0 \times e^{rt}\), where:

• \(N_t\) is the population size at time \(t\)
• \(N_0\) is the initial population size
• \(r\) is the growth rate
• \(t\) is time

In our task, we used this formula to arrive at \(N_t = 32 \times e^{0.5t}\). This equation shows us how the population size changes with time, given a starting population of 32 and a consistent growth rate of 0.5 per time unit. It offers a clear, concise method to estimate future population sizes under these conditions, making it invaluable for modeling biological growth, resource allocation, and ecological studies. Understanding each element of the formula helps predict how populations might expand in real-world scenarios.