Problem 123 - Algebra and Trigonometry Real Mathematics, Real People

Question

A landlocked lake has been selected to be stocked in the year 2015 with 5500 trout, and to be restocked each year thereafter with 500 trout. Each year the fish population declines \(25 \%\) due to harvesting and other natural causes. (a) Write a recursive sequence that gives the population \(p_{n}\) of trout in the lake in terms of the year \(n,\) with \(n=0\) corresponding to 2015. (b) Use the recursion formula from part (a) to find the numbers of trout in the lake for \(n=1,2,3\) and \(4 .\) Interpret these values in the context of the situation. (c) Use a graphing utility to find the number of trout in the lake as time passes infinitely. Explain your result.

Step-by-Step Solution

Answer

(a) The recursive sequence is \(p_{n+1} = (1 - 0.25)p_{n} + 500\), with \(n=0\) for 2015. (b) The number of trouts in the lake for \(n=1,2,3,4\) are 4625, 3973.75, 3480.3125, 3110.23437 respectively. (c) As time passes indefinitely, the number of trouts will stabilize around 2000.

Step 1: Formulate Recursive Sequence

The problem gives an initial condition for 2015 (or \(n = 0\)) as \(p_{0} = 5500\). Each year after that, 500 trouts are stocked, and the population reduces by \(25\%\). We can express this as a recursive sequence: \(p_{n+1} = (1 - 0.25)p_{n} + 500\)

Step 2: Compute Population for Years \(n=1,2,3,4\)

Using the recursive formula, for \(n=1,2,3,4\) we get:- \(p_{1} = 0.75 * p_{0} + 500 = 4625\)- \(p_{2} = 0.75 * p_{1} + 500 = 3973.75\)- \(p_{3} = 0.75 * p_{2} + 500 = 3480.3125\)- \(p_{4} = 0.75 * p_{3} + 500 = 3110.23437\)

Step 3: Analyse the Result for Infinite Time

For \(n = \infty\), we want to find the limit of this recursion, or when the population growth balances out the annual reduction. We set \(p_{n+1} = p_{n} = p\), we get:\(p = 0.75p + 500 \Rightarrow p = 2000\). Therefore, as time passes, the population will stabilize around 2000.

Key Concepts

Sequences and Series

Sequences and series are fundamental concepts in mathematics used to describe ordered lists of numbers. In this problem, we are dealing with a recursive sequence where each term depends on the previous term. Unlike arithmetic or geometric sequences, recursive sequences are defined by an initial value and a recurrence relation.
In this specific example, the initial value is the number of fish at the start, 5500, and the recurrence relation is given by the formula:

• \(p_{n+1} = 0.75p_{n} + 500\)

Here, each new population value is calculated by taking 75% of the current population and adding 500 fish that are restocked every year. This reflects both the natural decline due to harvesting and other factors, and the replenishment of the fish stock. The sequence, therefore, models a dynamic natural system which changes over time. Recursive sequences like these are versatile tools, useful in modeling various scenarios beyond fisheries, such as financial balances or population studies.

Mathematical Modeling

Mathematical modeling involves representing real-world situations through mathematical formulas and equations. In this exercise, we have a lake ecosystem being modeled with a recursive sequence. This model accounts for changes in trout population over time due to natural events and human intervention.
The initial condition sets the starting population. The relation accounts for a consistent decline of 25% because of harvesting effects and a consistent restocking effort. Mathematical models allow us to predict future outcomes.

• They simplify complex processes, allowing analysis and forecasting.
• The recursive sequence adapts easily to changed conditions by altering the parameters.

For instance, if the restocking number or decline rate changes, the model can quickly adjust to provide new predictions. Mathematical models like this one are powerful as they enable decision-makers to evaluate the potential long-term outcomes of different strategies.

Infinite Limits

Infinite limits often arise within sequences and series where the terms can approach a specific value. In this exercise, we're interested in what happens to the trout population as the number of years, \(n\), tends towards infinity.
By using the recursive formula, we find that the population eventually stabilizes at around 2000. This kind of behavior is a demonstration of a limit phenomenon. As time tends towards infinity, changes become negligible, resulting in a steady state.

• To find this limit, set \(p_{n+1} = p_{n} = p\) to solve for \(p\).
• This gives the stable population: \(p = 0.75p + 500\) leading to \(p = 2000\).

Understanding infinite limits is crucial because it tells us about the long-term behavior of the system. This aspect of mathematical analysis is used widely in fields like economics, physics, and biology, where modeling systems over extended periods can inform strategic decisions.