Problem 22 - Calculus for Biology and Medicine

Question

Find the exponential growth equation for a population that triples in size every unit of time and that has 72 individuals at time \(0 .\)

Step-by-Step Solution

Answer

The exponential growth equation is \( P(t) = 72 \cdot 3^t \).

Step 1: Understand the Exponential Growth Formula

Exponential growth can be modeled by the equation \( P(t) = P_0 \cdot a^t \) where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, and \( a \) is the growth factor.

Step 2: Identify the Initial Conditions

We're given that there are 72 individuals at time 0, which means \( P_0 = 72 \). Additionally, the population triples every unit of time, implying that the growth factor \( a = 3 \).

Step 3: Construct the Exponential Growth Equation

Substitute the values for \( P_0 \) and \( a \) into the exponential growth equation: \( P(t) = 72 \cdot 3^t \). This equation represents the population at any time \( t \).

Key Concepts

Population Growth

Population growth is a key biological and mathematical concept that explains how the number of individuals in a population increases over time. In certain cases, such as when resources are unlimited, a population might grow exponentially. This means it multiplies by a constant factor during each time period. This type of growth results in a rapid increase in population size. An evident example is the human population before resource limitations start affecting growth.

• In exponential growth, the population size doesn't just increase; it increases at a rate proportional to its current size.
• The formula for this type of growth is typically given as: \[ P(t) = P_0 \cdot a^t \]
• This equation shows that the population at any time, \( t \), depends on the initial population size and the factor by which it grows.

Understanding how populations grow with this model can help scientists and researchers make predictions about future population sizes. It also allows environmental planners to assess the impact of such growth on resources and ecosystems.

Initial Conditions

Initial conditions are critical when modeling exponential growth. They determine where the growth begins, setting the stage for all future calculations in the model. In the context of population growth, the initial condition often refers to the population size at time zero.

• The initial population size, denoted as \( P_0 \), is the starting point of the population.
• It's crucial because even in exponential growth, the outcome at any later time \( t \) heavily relies on this starting value.

In exercises like the one you are studying, knowing the initial number of individuals allows for the proper setup of the exponential growth equation. If we begin with 72 individuals, for example, this number is plugged into the growth equation to pave the way for finding how the population evolves over time.
Always keep in mind that while the initial condition might seem straightforward, it is fundamental to the accuracy and relevance of any predictive model.

Growth Factor

The growth factor is the multiplier that shows how much the population grows over each period of time. In exponential growth, this factor remains constant and is a crucial part of understanding how populations change.

• Within the equation \( P(t) = P_0 \cdot a^t \), \( a \) represents the growth factor.
• If a population triples every unit of time, as in your exercise problem, then the growth factor \( a \) is equal to 3.
• A larger growth factor means a faster growth rate and vice versa.

It's vital to understand this component because it directly influences the shape and steepness of the exponential growth curve. For instance, a growth factor of 3 indicates that the population is growing at a rapid rate, increasing threefold each time period.
Understanding the growth factor isn't just about predicting population sizes; it also helps contextualize biological, ecological, and environmental implications of such rapid population changes.