Problem 33 - Calculus Early Transcendentals

Question

Suppose that 100 fruit flies are placed in a breeding container that can support at most \(10,000\) flies. Assuming that the population grows exponentially at a rate of \(2 \%\) per day, how long will it take for the container to reach capacity?

Step-by-Step Solution

Answer

It takes approximately 230 days for the fruit flies to reach the container's capacity.

Step 1: Understand the Problem

We are given an initial population of 100 fruit flies and a container that can support up to 10,000 flies. The population grows at an exponential rate of 2% per day. We need to determine how many days it will take for the population to reach 10,000.

Step 2: Set Up the Exponential Growth Model

The formula for exponential growth is given by \[ P(t) = P_0 e^{rt} \]where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in days. Here, \( P_0 = 100 \), \( r = 0.02 \), and \( P(t) = 10,000 \).

Step 3: Solve for the Time \( t \)

Substitute the values into the growth model:\[ 10,000 = 100 e^{0.02t} \]Divide both sides by 100:\[ 100 = e^{0.02t} \]Take the natural logarithm of both sides to solve for \( t \):\[ \ln(100) = 0.02t \]\[ t = \frac{\ln(100)}{0.02} \]

Step 4: Calculate \( t \)

Compute the natural logarithm from the previous step:\[ \ln(100) \approx 4.605 \]Then divide by 0.02:\[ t = \frac{4.605}{0.02} \approx 230.25 \]Thus, it will take approximately 230.25 days.

Key Concepts

Population Growth

Population growth can vary depending on several factors, and one common model to describe it is exponential growth. In exponential growth, the population increases by a constant percentage over equal time intervals. This means that as the population gets larger, the amount it grows will also get larger due to the accumulation of the previous increases.
In our example with the fruit flies, we see a starting population of 100 and a daily growth rate of 2%. Use this information to estimate how the population will change over time. This kind of growth is not linear, where the increase is steady; instead, it accelerates as time goes on.

• Initially, the population grows slowly as each new cycle builds on a small starting number.
• Over time, the population will increase rapidly, especially as it approaches the maximum capacity of the environment.

Exponential growth can provide a simple yet powerful way to predict how populations may escalate and help illustrate the need for limits in resources and space.

Exponential Functions

Exponential functions are mathematical expressions used to model situations where growth or decay happens at a constant percentage rate over time. These functions are characterized by a variable exponent and can describe phenomena such as population growth, radioactive decay, and interest compounding.
In the formula for exponential growth, \[ P(t) = P_0 e^{rt} \]

• \( P(t) \) represents the population at time \( t \).
• \( P_0 \) is the initial population.
• \( r \) stands for the growth rate.
• \( t \) is the time period.

This equation is vital because it quantifies how populations can expand or contract over time when subjected to a specific rate. The " \( e \) " is the base of the natural logarithm, approximately equal to 2.718, which makes these calculations possible and consistent across different scientific applications.

Mathematical Modeling

Mathematical modeling is a crucial process that uses mathematical concepts and equations to simulate real-world situations. Whether in ecology, finance, or physics, it helps us predict outcomes and understand complex systems.
For our fruit fly population exercise, we used a mathematical model to determine how long it would take for the population to reach its capacity.

• The initial data, like population size and growth rate, are essential starting points.
• Models use these inputs to represent the behavior of a system over time.
• They help us visualize potential outcomes and plan for future scenarios.

By applying the exponential growth model, we could estimate that it would take approximately 230.25 days for the fruit fly population to reach the carrying capacity of their environment. This modeling process provides insights not just into the fruit fly population but can also apply to various other growth scenarios in real-world contexts.