Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Pest eradication In a pest eradication program, \(N\) sterilized male flies are released into the general population each day. It is estimated that \(90 \%\) of these flies will survive a given day. (a) Show that the number of sterilized flies in the population \(n\) days after the program has begun is $$ N+(0.9) N+(0.9)^{2} N+\cdots+(0.9)^{k-1} N $$ (b) If the long-range goal of the program is to keep 20,000 sterilized males in the population, how many flies should be released each day?

Step-by-Step Solution

Answer

Release 2,000 sterilized flies each day to maintain 20,000 in the population.

Step 1: Understanding the Problem Statement

We are given that each day, a certain number, \(N\), of sterilized male flies are released and that \(90\%\) of them survive each subsequent day. We need to calculate the total number of sterilized flies \(n\) days after release and determine how many should be released daily to maintain 20,000 in the population.

Step 2: Calculate the Number of Sterilized Flies After n Days

Each day \(k\), \(N\) new sterilized flies are added, and \(90\%\) of the previous day's flies survive. Therefore, the total number of surviving flies from \(N\) released days ago is \((0.9)^{k-1}N\). Summing this for all \(k\) days gives: \(N + (0.9)N + (0.9)^2N + \cdots + (0.9)^{k-1}N\). This series represents the number of flies on the \(n\)-th day.

Step 3: Understand the Infinite Geometric Series Formula

Recognize the series is a geometric progression with first term \(a = N\) and common ratio \(r = 0.9\). The sum of an infinite geometric series, \(S\), is given by \(S = \frac{a}{1-r}\) when \(|r| < 1\).

Step 4: Apply the Series Formula to Find N for 20,000 Flies

For the program to maintain 20,000 flies: \(\frac{N}{1-0.9} = 20,000\). Simplifying: \(\frac{N}{0.1} = 20,000\) which results in \(N = 20,000 \times 0.1\).

Step 5: Calculate the Daily Release Requirement

Solve for \(N\): \(N = 2,000\). Therefore, the program should release 2,000 sterilized flies daily to maintain the target of 20,000 flies in the population.

Key Concepts

Sterilized Flies Population

In the context of a pest eradication program, managing the population of sterilized flies plays a crucial role. Think of it this way: each day a certain number, \(N\), of sterilized male flies are introduced into the environment. These flies help control the pest population by interrupting the breeding cycle. However, not all sterilized flies survive daily. It's estimated that only 90% will make it to the next day.

The ongoing release and daily survival create a sequence where every day the surviving flies from the previous releases contribute to the total population. If you release \(N\) flies today, tomorrow, \(0.9N\) will likely still be there. Over time, the total number of sterilized flies in the general fly population is built up by this series of decaying amounts: \(N + 0.9N + 0.9^2N + \cdots\). Eventually, this sequence allows us to control the number of sterilized flies at any given time.

Pest Eradication Program

A pest eradication program is essentially an organized effort to reduce or eliminate pest populations. The one involving sterilized flies is an innovative method under biological control strategies. By releasing sterilized males into a pest population, you're targeting the core of reproduction.

• The sterilized flies compete with wild males for mates.
• Females that mate with sterilized males will not produce offspring.
• This gradually leads to a reduction in the pest population over time.

By recalibrating the number of sterilized flies released each day, the program can dynamically manage the population size. The primary goal is to reduce pest numbers significantly, or ideally, eradication in a certain period while ensuring a stable number of sterilized individuals are circulating within the population continually.
This method is environmentally friendly and offers an alternative to chemical pesticides, focusing instead on disrupting the breeding process rather than directly attempting to reduce the pest numbers through extermination.

Infinite Geometric Series Formula

The infinite geometric series formula is a powerful mathematical tool that deals with sums of sequences. Here, the series involved, \(N + (0.9)N + (0.9)^2N + \cdots\), is geometric because each term after the first is obtained by multiplying the previous term by a constant, which is the common ratio \(r=0.9\).
The sum of an infinite geometric series, when you have \(|r| < 1\), can be calculated using the formula:\[S = \frac{a}{1-r}\]where \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio.In the context of the pest eradication program, this formula helps calculate how many sterilized flies should be released (\(N\)) to have a stable population of 20,000 flies. Plugging values into the formula \( \frac{N}{0.1} = 20,000 \), we solve for \(N = 2,000\). Therefore, by releasing 2,000 flies per day, the program can maintain the desired population, leveraging the mathematical elegance of the infinite geometric series to inform practical ecological management.