Problem 39 - Advanced Engineering Mathematics

Question

Population Dynamics 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

The culture grows at a rate of 0.3 million cells per hour when it is 2 million cells.

Step 1: Understand the Problem

We are given a differential equation \( P'(t) = 0.15 P(t) \). This represents the rate of growth of the population of a cell culture, where \( P(t) \) denotes the population size at time \( t \). The question asks us to find the rate of growth (i.e., \( P'(t) \)) at the specific time \( t \) when the population size is 2 million cells.

Step 2: Identify Given Values

We know that \( P(t) = 2 \) million when calculating \( P'(t) \). Thus, the size of the culture at the specific time \( t \) is 2 million cells. We need to use this information to find \( P'(t) \).

Step 3: Substitute into the Differential Equation

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

Step 4: Calculate the Rate of Growth

Perform the multiplication to find \( P'(t) \):\[ P'(t) = 0.15 \times 2 = 0.3 \]

Step 5: Interpret the Result

The calculation gives us \( P'(t) = 0.3 \). This means that at the time \( t \) when the culture reaches 2 million cells, the culture is growing at a rate of 0.3 million cells per hour.

Key Concepts

Differential Equations

Differential equations are mathematical tools used to model systems that change over time. At their core, they describe how a particular quantity, such as population size, evolves. In the context of population dynamics, differential equations are powerful because they provide insights into growth patterns.

• The differential equation given, \( P'(t) = 0.15 P(t) \), tells us how the population grows at any moment \( t \). Here, \( P'(t) \) is the rate of change of the population over time.
• This specific equation is a first-order linear differential equation, showing that the rate of growth is proportional to the current population size.

By modeling with differential equations, scientists can predict how populations such as cell cultures are expected to grow or shrink under certain conditions. They help in understanding complex biological processes, making them an essential tool in fields like biology and ecology.

Growth Rate

In population dynamics, understanding the growth rate is crucial for analyzing how quickly a population expands. The growth rate can vary depending on numerous factors, like resources or environment.

• In our model, the growth rate is encapsulated in the constant 0.15, which is multiplied by the current population size \( P(t) \).
• This constant represents a 15% growth per hour, suggesting that the cell culture will increase by 15% every hour.

Knowing the growth rate allows prediction of future population sizes. It helps in planning for resource allocation and environmental impacts, and gives critical insights into how populations respond to changes over time.

Cell Culture Growth

Cell culture growth refers to the process of cells growing and dividing in a controlled environment. This is vital in fields like medical research and pharmaceuticals.

• The cell culture growth is often modeled to understand how cells multiply. The equation \( P'(t) = 0.15 P(t) \) is a simple yet effective representation of this process.
• In the exercise, when the population size is 2 million cells, the rate \( P'(t) = 0.3 \) stands for an increase of 0.3 million cells per hour.

Through such mathematical models, researchers can simulate and predict the behavior of cell cultures. This understanding helps in developing new treatments and understanding diseases, making this growth modeling highly significant in biological and medical sciences.