Problem 21 - 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 20 individuals at time \(0 .\)

Step-by-Step Solution

Answer

The exponential growth equation is \( P(t) = 20 \times 3^t \).

Step 1: Understanding the General Exponential Growth Model

An exponential growth model can be expressed by the equation \[ P(t) = P_0 imes a^t \]where:- \( P(t) \) is the population at time \( t \).- \( P_0 \) is the initial population size.- \( a \) is the growth factor.- \( t \) is the time in consistent units. In this problem, \( P_0 = 20 \) is given.

Step 2: Defining the Growth Factor

The population triples every unit of time. This means the growth factor \( a \) is 3. Using this information, the equation simplifies to \[ P(t) = 20 imes 3^t \].

Step 3: Writing the Final Exponential Growth Equation

Now that we have defined both the initial population and the growth factor, we can write the final equation for the exponential growth of the population:\[ P(t) = 20 imes 3^t \].This equation models the population, which starts at 20 and triples every unit of time.

Key Concepts

Population Model

An exponential population model is a mathematical representation used to predict how populations grow over time. The essence of this model is to structure the growth process using a pattern that repeats over equal time intervals. This kind of growth is depicted using the exponential growth model equation:\[P(t) = P_0 \times a^t\]Here’s what this means:- **Population at Time t (\(P(t)\))**: This is the number of individuals in the population at a specific time \(t\).- **Consistent Timing**: In exponential models, time should be measured in consistent and uniform units, like days or years.The exponential population model offers a simple yet powerful way to understand and predict population dynamics. While the real world might be more complex, this model helps lay down a foundational understanding of growth patterns.Understanding a basic population model helps us see how rapidly or slowly a population can expand given a specific growth factor. The model helps demonstrate the potential impact of a small growth factor over time and accentuates the importance of timing.

Growth Factor

The growth factor is the core component of an exponential growth model. It dictates how the population changes with each passing unit of time. In the context of exponential growth, the growth factor is denoted by \(a\) in the model equation. - **Multiplier Effect**: If the growth factor is exactly 1, the population remains constant. A factor greater than 1 signals growth, while less than 1 shows a decline.- **Tripling Example**: In this exercise, it’s given that the population triples each time period, so \(a = 3\). This means that every given unit of time, the population size multiplies by 3.Using a growth factor lets us easily calculate the future size of a population if the rate consistently remains the same. The simplicity of the growth factor doesn’t take away from its power in predicting outcomes over numerous time units. Multiply that consistent factor over several periods, and you begin to see exponential expansion.

Initial Population Size

Initial population size, denoted as \(P_0\) in exponential growth models, sets the starting point for growth calculations. It is the population size from which growth begins.- **Starting Point Importance**: If \(P_0\) is underestimated, predictions will undervalue future population size.- **Example Context**: In our exercise, \(P_0 = 20\). This means the calculations for future growth using our model begin with a base of 20 individuals.There's always a seed or a beginning. Whether it's bacteria in a petri dish or a human population in a city, this initial number is crucial. It allows accurate modeling and realistic forecasts as it forms the baseline for future predictions.Even when populations double or triple over time with a growth factor, understanding and accurately establishing the starting point ensures that projections made by the exponential growth model remain grounded.