Problem 10 - Calculus for Biology and Medicine

Question

In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. Suppose you measure the following data for the size of a population of bacteria: $$ \begin{array}{lcccc} \hline \boldsymbol{t} & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{N}_{t} & 10 & 30 & 90 & 270 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The formula for the population is \(N(t) = 10 \times 3^t\).

Step 1: Observe the Change in Population

Notice how the population changes from one period to the next: \(N_1 = 3 \times N_0\), \(N_2 = 3 \times N_1\), and \(N_3 = 3 \times N_2\). This suggests a pattern where each term is a multiple of the previous term by 3.

Step 2: Identify the Pattern

The population seems to follow an exponential growth where each term is obtained by multiplying the previous term by 3. Thus, the pattern is \(N_t = 10 \times 3^t\), since \(10\) is the population at \(t = 0\) and every subsequent term is multiplied by 3.

Step 3: Generalize the Pattern

From the observed pattern, we can conclude that the population at any time \(t\) can be described by the formula \(N(t) = 10 \times 3^t\). This formula accounts for the initial population and the constant growth factor.

Key Concepts

Understanding Bacterial Population

Bacterial populations are an excellent example of exponential growth in biology. They can increase rapidly under the right conditions due to their quick reproduction cycles. In the context of a lab, monitoring bacterial growth helps scientists understand how bacteria might behave in different environments. In this problem, we discuss a scenario where a bacterial population is measured at different time points, and the data follows a particular growth pattern. At the start, or time zero (\(t=0\), the population is recorded as 10. With time, this bacterium seemingly triples every time unit, resulting in a dramatic increase in population. This kind of growth is common in bacteria due to their ability to reproduce rapidly. Understanding these populations is essential, not only for research but also for applied fields like medicine, where controlling bacterial growth is crucial.

Decoding the Growth Factor

The term *growth factor* in the context of bacterial growth refers to the number by which the population multiplies over a set time period. It describes the exponential increase of the population. In our exercise, the growth factor is 3. This means that for every time unit passed, the bacterial population triples. For instance:

• The population at \(t = 0\) is 10.
• At \(t = 1\), it becomes 30 (10 times 3).
• And by \(t = 2\), it reaches 90 (30 times 3).

When solving real-life problems, the growth factor helps in predicting how large a population might grow over time. It is a key part of the formula in our exercise, allowing us to mathematically express the exponential nature of the population increase.

Exploring Mathematical Modeling

Mathematical modeling is a powerful tool used to describe and predict phenomena. In this problem, the population growth model helps us describe how the number of bacteria changes over time. This specific model can be expressed using the equation: \[N(t) = 10 \times 3^t\] This equation captures the *initial population* of 10 and applies the *growth factor* of 3 over time \(t\). Each part of the model tells us something essential:

• \(10\): the number of bacteria we start with.
• \(3\): the multiplication rate or growth factor.
• \(^t\): time in discrete steps or intervals.

With this model, scientists and mathematicians can extrapolate data to predict where future values lie, determine bacterial load at any given time, and plan interventions if needed. Mathematical models like these simplify complex biological systems, making them more accessible for analysis and simulation.