Problem 24 - Calculus for Biology and Medicine

Question

Find the exponential growth equation for a population that quadruples in size every unit of time and that has 17 individuals at time 0 .

Step-by-Step Solution

Answer

The exponential growth equation is \( P(t) = 17 \times 4^t \).

Step 1: Understand the Problem

We are given that a population grows exponentially by quadrupling every unit of time. At time 0, there are 17 individuals. We need to find the formula for this exponential growth.

Step 2: Identify the Formula

The general formula for exponential growth is given by \( P(t) = P_0 imes b^t \), where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, and \( b \) is the growth factor per time unit.

Step 3: Determine Initial Population

The initial population \( P_0 \) at time \( t = 0 \) is 17 individuals. So, \( P_0 = 17 \).

Step 4: Determine the Growth Factor

Since the population quadruples every unit of time, the growth factor \( b \) is 4. This means that every time period, the population is multiplied by 4.

Step 5: Write the Exponential Growth Equation

Substituting the initial population and growth factor into the formula: \[ P(t) = 17 \times 4^t \] This equation represents the population at any time \( t \).

Key Concepts

Population Growth

Population growth refers to the change in the number of individuals in a population over time. It can occur through various mechanisms, such as births, deaths, immigration, and emigration. In our context of exponential growth, population growth is described as a steady increase at a fixed rate per unit of time. Imagine a scenario where a single organism multiplies at a rate such that its numbers steadily increase by a set multiple during consistent intervals.
In exponential population growth:

• We start with an initial population size (known as the initial condition).
• The size of the population increases by a constant ratio from one period to the next.
• The growth process does not take into account external factors like limited resources, which often affects real-world populations.

Understanding these dynamics is crucial when predicting future population sizes in scenarios like ecological studies or resource management. In our example, we know the initial population is 17, and it will grow by quadrupling every time period.

Growth Factor

The growth factor in an exponential growth model is the constant multiplier that determines how much the population will increase in each time period. This factor is crucial as it reflects the rate of change in population size.
To calculate the growth factor, you ask, "By what factor does the population increase each unit of time?" In our problem, the population quadruples (multiplied by 4) per time unit. Therefore, the growth factor is 4.

• A growth factor of 4 implies that if there are 17 individuals at time 0, there will be 68 individuals after one time unit.
• The understanding of the growth factor allows one to predict how quickly a population can reach a certain size.

In real-world applications, different organisms and their environments cause growth factors to vary widely, making it important to identify this factor accurately in studies of population dynamics.

Exponential Equation

An exponential equation for population growth provides a mathematical model that describes how a population increases over time. It can be depicted with the formula: \[ P(t) = P_0 \times b^t \]In this formula, \( P(t) \) represents the population at a given time \( t \), \( P_0 \) is the initial population size, and \( b \) is the growth factor. Plugging in our numbers, we get:\[ P(t) = 17 \times 4^t \]This equation tells you that starting from 17 individuals, the population will multiply by 4 for every time unit elapsed. For example:

• At \( t = 1 \), \( P(1) = 17 \times 4^1 = 68 \).
• At \( t = 2 \), \( P(2) = 17 \times 4^2 = 272 \).

The power of exponential equations lies in their ability to model how quickly populations can grow under ideal conditions. They highlight how small initial populations can become very large over time when the growth factor is substantial, as demonstrated in our exercise.