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Chapter 2: Problem 32

Let \(P(t)\) be the population of a bacteria culture after days and suppose that \(P(t)\) has the line \(y=25,000,000\) as an asymptote. What does this imply about the size of the population?

Short Answer

Expert verified
The population of the bacteria culture will approach and stabilize at 25,000,000 over time.

Step by step solution

01

Understand the Problem

The exercise asks to determine the implication of the population function of bacteria, denoted as \(P(t)\), having \(y=25,000,000\) as an asymptote. An asymptote is a line that a function approaches as the input (in this case, time \(t\)) goes to infinity.
02

Identify the Asymptote

The line given is \(y=25,000,000\). This line acts as an asymptote for the function \(P(t)\). It implies that as time \(t\) increases, the value of \(P(t)\) gets closer and closer to this line
03

Interpret the Asymptote

Since \(y=25,000,000\) is a horizontal asymptote for \(P(t)\), it means that the population \(P(t)\) of the bacterial culture is approaching 25,000,000 as the number of days \(t\) goes to infinity. In biological terms, this suggests a maximum sustainable population limit.
04

Conclude the Implication

Therefore, the bacteria population will grow over time but will eventually level off and stabilize, approaching a maximum size of 25,000,000 bacteria. The population will not exceed this number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

population function
In this context, the population function, denoted as \( P(t) \), represents how the population of a bacteria culture changes over time. Here, \( t \) stands for time measured in days. The function provides a way to understand how the bacterial population grows and evolves. In many cases, especially with natural populations, this growth is not unlimited. Various factors such as resources and space can limit growth. The population function captures all these dynamics in a mathematical form. By analyzing \( P(t) \), we can predict future population sizes and understand how the population behaves over time.
horizontal asymptote
A horizontal asymptote of a function is a horizontal line that the function approaches as the input goes to infinity. In the problem, the line \( y = 25,000,000 \) is identified as the horizontal asymptote for the population function \( P(t) \). This means that as time \( t \) progresses, the population \( P(t) \) gets very close to 25,000,000 but never quite reaches or exceeds this number.
This behavior is significant because it indicates a limiting behavior in the population growth. No matter how much time passes, the bacterial population will not grow beyond 25,000,000. Instead, it will stabilize around this value. This concept helps in understanding long-term population trends and potential resource consumption limits.
maximum sustainable population limit
The concept of a maximum sustainable population limit is directly related to the horizontal asymptote discussed earlier. This limit, in the context of the given problem, is 25,000,000. The horizontal asymptote \( y = 25,000,000 \) implies that the is the maximum number of bacteria that the environment can support indefinitely.
This idea is crucial in biological and environmental science because it considers the capacity of a habitat to support a population over a long period without degrading. Factors like nutrients, space, and waste accumulation influence this limit. For the bacteria culture, a maximum sustainable population limit of 25,000,000 means that there are enough resources to keep the population stable at that number. If the population were to grow beyond this size, it could lead to resource depletion and a subsequent decline in population.

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