Problem 20 - 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 that triples in size every unit of time and that has 72 individuals at time 0 .

Step-by-Step Solution

Answer

The formula is \(N_t = 72 \cdot 3^t\).

Step 1: Understand the Problem

We need to find a formula of population size \(N_t\) as a function of time \(t\). The population triples every unit time and starts with 72 individuals at \(t = 0\).

Step 2: Exponential Growth Formula

The general formula for exponential growth is \(N_t = N_0 \cdot a^t\), where \(N_0\) is the initial population, \(a\) is the growth factor, and \(t\) is time.

Step 3: Determine Growth Factor

Since the population triples every unit of time, the growth factor \(a = 3\). Therefore, the proper growth factor in our formula is \(3\).

Step 4: Write the Specific Formula

Substitute the initial population \(N_0 = 72\) and the growth factor \(a = 3\) into the formula from Step 2: \[N_t = 72 \cdot 3^t\].

Step 5: Verify the Formula

Check that \(N_0 = 72\) at \(t = 0\). When \(t = 0\), \(N_t = 72 \cdot 3^0 = 72 \cdot 1 = 72\), confirming the initial population is correct.

Key Concepts

Population Size Formula

When dealing with exponential growth scenarios, the population size formula is fundamental. This formula helps us model how populations change over time. It goes like this: the population at time \( t \), noted as \( N_t \), is equal to the initial population \( N_0 \) multiplied by the growth factor \( a \), raised to the power of the time \( t \). It is written as:\[N_t = N_0 \cdot a^t\]

• \( N_0 \): the initial population size, which is where it all begins, at time \( t = 0 \).
• \( a \): the growth factor that influences how much the population increases every time interval.
• \( t \): the time that has passed since the beginning of the observation.

Understanding this structure is key, as it tells us how populations grow exponentially, with each unit of time causing the population to increase by the same percentage.

Growth Factor

The growth factor is crucial to understanding exponential growth. It represents how many times the population multiplies over a single time unit. In our problem, we've learned that the population triples every unit of time. This means if you have 72 individuals to start, one time unit later you will have three times that amount.In formula terms, the growth factor \( a \) is 3. This means at each time step:

• The population isn't just increasing by the same number each time; it's multiplying.
• Because the growth is exponential, each passing time unit sees the population grow based on its current size, multiplying by 3.

The growth factor is the lever that dictates just how rapid or modest the growth rate is.

Initial Population

The initial population \( N_0 \) is the starting size of the population we're examining. It's the number of individuals you have before any time has passed, specifically at \( t = 0 \). This value serves as the baseline in our exponential growth formula.In the given exercise, we're informed that the initial population is 72 individuals. This number is pivotal because it gives us a starting point to apply the exponential growth process.

• Without knowing \( N_0 \), we couldn't calculate the future population sizes accurately.
• Each subsequent population size is a reflection of not only the growth factor but also this initial figure.

In our case, all future calculations of \( N_t \) depend on this initial population being correctly accounted for, highlighting its importance in long-term projections.