Problem 5 - Calculus for Biology and Medicine

Question

\(5-10\), give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=2\); population doubles every 20 minutes; one unit of time is 20 minutes

Step-by-Step Solution

Answer

The population formula is \(N(t) = 2^{t+1}\). Validating confirms it accurately models population doubling every 20 minutes.

Step 1: Understand the Problem

We are given an initial population size and an instruction that the population doubles every 20 minutes. Our task is to find a formula for the population at any time \(t\), where \( t \) is measured in 20-minute intervals.

Step 2: Identify the Initial Values

We know from the problem statement that \(N_0 = 2\). This means that at time \(t = 0\), the population size is 2.

Step 3: Determine the Growth Pattern

Since the population doubles every 20 minutes, the population at time \(t = 1\) will be \(2\times2 = 4\), at \(t = 2\) it will be \(4\times2 = 8\), and so on. Observing this pattern shows that the growth is exponential.

Step 4: Derive the Population Formula

Given the doubling pattern, the general formula for the population at any time \(t\) is \(N(t) = N_0 \times 2^t\). We plug in the initial population value \(N_0 = 2\) to get: \[N(t) = 2 \times 2^t = 2^{t+1}\]

Step 5: Validate the Formula

Let's double-check: At \(t = 0\), \(N(0) = 2^{1} = 2\). At \(t = 1\), \(N(1) = 2^{2} = 4\), and at \(t = 2\), \(N(2) = 2^{3} = 8\). The formula matches the expected values, confirming its correctness.

Key Concepts

Understanding Initial Population

In the context of exponential growth, the initial population is crucial. It represents the starting point from which growth begins. Here, it is denoted by \( N_0 \). When given a problem related to population growth, the first step is to identify this initial value.

In our exercise, the initial population \( N_0 \) is provided as 2. This means that at the very beginning, or at time \( t = 0 \), there are 2 units of whatever population we are considering, say bacteria or cells. This value is fixed and does not change; it serves as the baseline for calculating future population sizes.

Remember, the initial population is key for constructing the formula that will predict future populations. Without it, it would be impossible to determine the starting point for any calculations or further projections.

Decoding Doubling Time

The concept of doubling time is central to understanding exponential growth. It tells us how long it takes for a population to double in size. This period is a constant in exponential growth problems.

In the exercise, the doubling time is given as 20 minutes. This means every 20 minutes, the population size grows twice as large as it was at the start of that period. Knowing the doubling time helps us establish the rate at which growth occurs.

When setting up the timeline of growth, it's important to track each interval of time as it represents a complete doubling cycle. This consistent, predictable increase is why doubling time is a valuable tool for projecting future population sizes. There is predictability in knowing exactly when the size will double, making it easier to apply in applications ranging from biology to finance.

Crafting the Population Formula

Having established the initial population and knowing the doubling time enables the creation of a population formula. This formula is a mathematical representation that allows us to predict the population size at any given time.

In this exercise, the formula we derive is formulated as \( N(t) = 2^{t+1} \). Let's break it down:

• \( N_0 = 2 \), which is the initial population.
• \( 2^t \) represents the doubling pattern over time \( t \), as each unit of \( t \) corresponds to a complete doubling cycle (and since it doubles every 20 minutes, \( t \) is in intervals of 20 minutes).
• The formula captures how the population expands exponentially, doubling with each increment of \( t \).

Using the formula is straightforward once it's set up. To find the population at any time \( t \), simply plug \( t \) into the equation. Efficient adherence to this formula means you can calculate future populations with ease and accuracy.