Problem 67 - Calculus: Early Transcendentals

Question

A fish has 5000 catfish in his pond. The number of catfish by \( 8 \% \) per month and the farmer harvests 300 catfish per month. (a) Show that the catfish population \( P_n \) after \( n \) months is given recursively by \( P_n = 1.08 P_{n-1} - 300\) \(P_0 = 5000\) (b) How many catfish are in the pond after six months?

Step-by-Step Solution

Answer

After 6 months, the pond has 5735 catfish.

Step 1: Understand the Problem

We have a pond with 5000 catfish initially. Each month, the number of catfish increases by \(8\%\) and the farmer removes 300 catfish from the pond.

Step 2: Formulate the Recursive Formula

The catfish population grows by \(8\%\), meaning each month the population from the previous month is multiplied by \(1.08\). After the growth, the farmer harvests 300 catfish. This creates the recursive formula: \[ P_n = 1.08 P_{n-1} - 300 \].

Step 3: Check Initial Condition

We know the initial number of catfish, \(P_0 = 5000\). This serves as the starting point for our recursive formula.

Step 4: Calculate Population After First Month

Using the recursive formula: \[ P_1 = 1.08 imes 5000 - 300 = 5400 - 300 = 5100\].

Step 5: Calculate Population After Second Month

Continue using the recursive formula:\[ P_2 = 1.08 imes 5100 - 300 = 5508 - 300 = 5208\].

Step 6: Calculate Population After Third Month

Calculate using:\[ P_3 = 1.08 imes 5208 - 300 = 5624.64 - 300 = 5324.64\]. Since we deal with whole catfish, we can round to \(5325\).

Step 7: Calculate Population After Fourth Month

Find the population with:\[ P_4 = 1.08 imes 5325 - 300 = 5751 - 300 = 5451\].

Step 8: Calculate Population After Fifth Month

Proceed with:\[ P_5 = 1.08 imes 5451 - 300 = 5887.08 - 300 = 5587.08\]. Round to \(5587\).

Step 9: Calculate Population After Sixth Month

Finally, using the formula:\[ P_6 = 1.08 imes 5587 - 300 = 6034.96 - 300 = 5734.96\]. Round to \(5735\).

Step 10: Complete the Solution

After calculating step by step with the recursive formula, we find that the number of catfish in the pond after six months is 5735.

Key Concepts

Population Modeling

In this exercise, we are dealing with population modeling to figure out how the number of catfish changes over time in a controlled environment. Population modeling is a method used to understand how populations, like fish, grow or shrink over time based on specific influences or controls, such as growth rates and harvesting rates. This type of modeling is helpful for numerous fields including ecology, environmental management, and even economics.

We start with an initial population of 5000 catfish. The population grows by a certain percentage each month, here it is an 8% increase. This means the catfish population is not static, and without any intervention, it would continue to swell.

• The 8% growth is a natural process where each month's population is 1.08 times that of the previous month.
• The harvesting of 300 catfish per month counteracts this growth, indicating human intervention to control the population size.

This scenario is a practical application of a balanced population model, where one factor (growth) is offset by another (harvesting). It's vital to have such models to predict future population sizes and to manage natural resources effectively.

Discrete Mathematics

Discrete mathematics involves studying distinct and separated objects, and recursive sequences, like the one in our exercise, are a pivotal part of this branch. Recursive sequences define a series where each term is determined by the preceding ones, using a set mathematical operation. In our case, we define each monthly fish population based on the previous month's population.

Let's explore the components in more detail:

• The recursive formula we've used is: \[ P_n = 1.08 P_{n-1} - 300 \]. This identifies that the population at any month 'n' depends on the population of the previous month 'n-1'.
• The initial condition, given by \( P_0 = 5000 \), sets the starting point for calculating future populations.

Through recursion, we can model complex systems, predict behaviors, and analyze stability in populations. This method is crucial in understanding the step-by-step process of population dynamics in our problem.

Calculus

Although primarily this problem operates within the realm of discrete mathematics, calculus also provides significant insights into population dynamics, particularly when exploring continuous growth models. Here, we use discrete steps (each month) to calculate growth, while calculus could help in understanding the continuous change.

How might calculus be applied to such a problem?

• **Continuous Growth:** By extending calculus concepts, we can consider how populations grow continuously, which would involve using differential equations to model these changes.
• **Growth Rates:** Calculus helps us grasp the concept of growth rates (in this case, 8%), showing how these rates impact the overall change over time.

While our catfish problem uses a discrete approach because the changes occur monthly, understanding calculus can give advanced students a further edge in exploring more dynamic models of population change, especially when translating to larger datasets or more complex ecosystems.