Problem 73 - Precalculus: Mathematical for Calculus

Question

A fish farmer has 5000 catfish in his pond. The number of catfish increases 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 \(P_{0}=5000\) and $$P_{n}=1.08 P_{n-1}-300$$ (b) How many fish are in the pond after 12 months?

Step-by-Step Solution

Answer

The recursive formula models the situation correctly, and after 12 months, there are approximately 6805 catfish.

Step 1: Understand the Problem

We start with 5000 catfish. Each month, the catfish population increases by 8%, and 300 catfish are harvested. We need to write a recursive formula for the population over months.

Step 2: Set Up Initial Condition

We identify the initial quantity of catfish in the pond. This is given as:\( P_0 = 5000 \). This is our starting point before any changes have occurred.

Step 3: Apply Population Growth

Each month, the population increases by 8%. If the current population is \( P_{n-1} \), the population before harvesting is:\( P_{n-1} \times 1.08 \).

Step 4: Account for Harvesting

After the growth, 300 catfish are harvested, reducing the population. The formula for the new population \( P_n \) becomes:\[ P_n = 1.08 P_{n-1} - 300 \]. This reflects both the growth and the harvest.

Step 5: Verify Recursive Formula

The recursive formula \( P_n = 1.08 P_{n-1} - 300 \) satisfies the conditions of 8% growth followed by a harvest of 300 catfish each month. We have shown that this correctly models the situation.

Step 6: Calculate Population After 12 Months

To find \( P_{12} \), we iterate using the recurrence relation starting from \( P_0 = 5000 \). Calculate each month up to \( P_{12} \) using the formula \( P_n = 1.08 P_{n-1} - 300 \).

Step 7: Perform Iterative Calculation

Calculate successive months:- \( P_1 = 1.08 \times 5000 - 300 = 5100 - 300 = 4800 \)- \( P_2 = 1.08 \times 4800 - 300 = 5184 - 300 = 4884 \)- Continue this process to find each month's population up to \( n = 12 \).

Step 8: Calculate P12

Continue the iteration, using the previous month's output each time, until you reach \( n = 12 \). For \( P_{12} \), repeat these calculations with each subsequent population until complete. This manual calculation is cumbersome if not using a calculator, ensure final value by double-checking arithmetic.

Key Concepts

Population Growth

In the context of our fish pond, population growth describes how the number of catfish changes over time. Starting with 5000 catfish, the population increases by 8% each month. This percentage represents the growth rate, which helps us calculate the new amount of catfish before any are harvested.
The growth rate is essential when observing how populations of living organisms, like fish, increase over time. It's a realistic factor as animals reproduce and grow naturally. The 8% rate means the population is multiplying by 1.08 each month.
Understanding population growth is valuable not only in biology but also in economics and demographics where similar principles apply. This growth is typically exponential, meaning it compounds over time, leading to faster increases as the base number itself increases. Incorporating realism, this exponential growth must also consider limits like resources and environmental constraints.

Recursive Formula

A recursive formula helps us describe a sequence where each term depends on the one before it. In our example with catfish, the formula is given as \( P_n = 1.08 P_{n-1} - 300 \). This means that to find the population in the current month \( P_n \), we use the previous month's population \( P_{n-1} \).
The recursive element lies in the fact that each output feeds into the next calculation. Initially, \( P_0 = 5000 \) provides our starting point. From there, the formula accounts for growth and harvesting systematically.
This structure is effective in scenarios where repeated processes occur, like our monthly fish stock changes. Recursive formulas simplify calculations for sequential situations and can be applied through straightforward arithmetic steps, pushing forward towards a desired result over a set period.

Arithmetic Calculations

Within our fish scenario, arithmetic calculations are pivotal. We move step by step through each month's population determination. The arithmetic involves applying multiplication first for the growth factor (1.08), then a subtraction representing the harvest.
Let's break down a single month's calculation. Suppose you start with 5000 catfish. First, multiply by 1.08: resulting in 5400. Then subtract the 300 harvested catfish, leaving a new count of 5100 for the following month.
- Example re-iteration: * Calculate growth: \( P_{1} = 1.08 \times 5000 = 5400 \) * Subtract harvest: \( 5400 - 300 = 5100 \)Repeating these arithmetic steps from \( n = 1 \) through \( n = 12 \) enables us to track changes over time systematically, ensuring accuracy in our final population count.