Problem 8 - Calculus for Biology and Medicine

Question

give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=6\); population doubles every 40 minutes; one unit of time is 60 minutes

Step-by-Step Solution

Answer

The formula for the population is \(N(t) = 6 \times (2^{1.5})^t\).

Step 1: Understand the Initial Conditions

The initial population at time \(t = 0\) is denoted as \(N_0\) and given as 6.

Step 2: Determine the Doubling Time in the Given Units

The population doubles every 40 minutes, but time \(t\) is given in 60-minute periods. To adjust for this, note that 60 minutes includes \(\frac{60}{40} = 1.5\) doubling periods.

Step 3: Establish the Doubling Effect per Time Unit

Every 60 minutes (which is one time unit \(t\)), the population doubles 1.5 times. This can be expressed as the population being multiplied by \(2^{1.5}\) for each unit time \(t\).

Step 4: Establish the Formula for the Population at Time t

The population at time \(t\) can be given by the formula \(N(t) = N_0 \times (2^{1.5})^t\). Substitute \(N_0 = 6\): \[ N(t) = 6 \times (2^{1.5})^t \].

Key Concepts

Population Doubling Time

Population doubling time is an important concept in exponential growth, particularly when studying populations. It refers to the time it takes for a population to double in size. In environments with abundant resources, populations grow exponentially, meaning they increase by a constant factor over equal time intervals.

In the exercise, the initial population is given as 6, and the population doubles every 40 minutes. However, the challenge arises when combining this with time units of 60 minutes. This requires recalculating how often the population doubles in each 60-minute interval. Thus, we discover that there are 1.5 doubling periods within every 60 minutes.

Understanding population doubling time is crucial for predicting future population sizes and understanding growth rates. It provides insight into how quickly a population can grow under ideal conditions, and is widely used in fields like ecology, microbiology, and demography.

Biological Population Models

Biological population models play a critical role in understanding how populations change over time. These models use mathematical equations to describe how populations grow, typically taking into account factors like initial size, growth rates, and carrying capacities.

In this exercise, we are dealing with exponential growth, which is a simple form of a biological population model. It assumes that the population size grows at a constant rate, doubling at regular intervals. The formula derived, \( N(t) = 6 \times (2^{1.5})^t \), captures the idea that the population size at any given time is directly proportional to the initial population size adjusted by the growth factor.

Biological population models help researchers make predictions and formulate conservation strategies. They also assist in identifying trends and potential issues in population dynamics, enabling more informed decision-making in managing natural resources and ecosystems.

Discrete Time Units

When modeling population growth, discrete time units often simplify calculations and conceptual understanding. They allow the measurement of changes over set intervals, such as days, months, or, in this exercise, 60-minute periods.

The benefit of using discrete time units lies in their ability to clearly illustrate population changes at each step. Here, since each unit represents an hour and the population increases by a factor of \( 2^{1.5} \) within each, it becomes straightforward to predict and analyze population trends.

This approach contrasts with continuous models, where changes are observed instantaneously. Using discrete time scales aids in aligning with practical scenarios where data is collected at regular intervals. Understanding this concept assists students and practitioners in efficiently analyzing growth patterns and making projections based on periodic data collection.