Problem 39 - Advanced Engineering Mathematics

Question

Suppose that \(P^{\prime}(t)=0.15 P(t)\) represents a mathematical model for the growth of a certain cell culture, where \(P(t)\) is the size of the culture (measured in millions of cells) at time \(t\) (measured in hours). How fast is the culture growing at the time \(t\) when the size of the culture reaches 2 million cells?

Step-by-Step Solution

Answer

0.30 million cells per hour.

Step 1: Understand the Problem

We are given the rate of change of a cell culture population as \( P'(t) = 0.15 P(t) \). We need to find out how fast the culture is growing when \( P(t) = 2 \) million cells. This means we're asked to find \( P'(t) \) at that point.

Step 2: Substitute the Given Size into the Rate of Change Equation

Substitute \( P(t) = 2 \) million cells into the equation \( P'(t) = 0.15 P(t) \). This gives us:\[P'(t) = 0.15 \times 2\]

Step 3: Calculate the Rate of Change

Perform the multiplication to find \( P'(t) \) when the culture size is 2 million cells:\[P'(t) = 0.30\]This means the culture is growing at a rate of 0.30 million cells per hour when the population is 2 million cells.

Key Concepts

Rate of Change

In the context of differential equations, the concept of the "rate of change" is crucial. It refers to how a quantity changes in relation to another. Here, we're looking at how the population of cells in a culture changes with time. The given equation, \( P'(t) = 0.15 P(t) \), explicitly tells us this rate of change. More specifically, it states that the rate at which the culture is growing, \( P'(t) \), is equal to 15% of the current population \( P(t) \). This is a differential equation expressing a proportional relationship between the rate of growth and the current population size.

• This model is common in scenarios where growth depends on current population size.
• The larger the population, the faster it grows.
• Finding \( P'(t) \) reflects the immediate growth rate at any given time.

When the population hits 2 million cells, this framework allows us to calculate its growth. We substitute the size into the differential equation to find that \( P'(t) = 0.30 \), indicating a growth of 0.30 million cells per hour.

Exponential Growth Model

The equation \( P'(t) = 0.15 P(t) \) is a classic example of an exponential growth model. This model is characterized by growth that is proportional to the current size. In other words, as the population increases, the absolute growth per unit time also increases, which results in an exponential curve when plotted over time. The proportional constant 0.15 is key because it suggests that the culture grows by 15% every hour.

• This type of model is frequently seen in populations without limiting factors, such as unlimited resources or space.
• Exponential growth can lead to very rapid increases in population size over time.
• It's a simple yet powerful way to predict future population sizes, especially when conditions remain constant.

Understanding exponential growth helps predict how quickly a small culture might grow into a very large one under ideal conditions.

Applications in Biology

Differential equations and exponential growth models are not just mathematical concepts; they are instrumental in biology. They help describe how organisms, from bacteria to mammalian cells, multiply over time. This particular model can apply to cultures grown in controlled environments, like petri dishes, where resources are initially abundant.

• This model explains bacterial growth phases where they double in number regularly.
• It's used to predict the growth of cancer cells or the spread of infectious diseases.
• Beyond cells, these models apply to population dynamics in ecology.

By employing such models, biologists can foresee changes in ecosystems or optimize conditions for cell culture growth.