Problem 82 - Algebra and Trigonometry

Question

Trout Population A pond currently contains 2000 trout. A fish hatchery decides to add 20 trout each month. It is also known that the trout population is growing at a rate of \(3 \%\) per month. The size of the population after \(n\) months is given by the recursively defined sequence \(p_{0}=2000 \quad p_{n}=1.03 p_{n-1}+20\)

Step-by-Step Solution

Answer

Use \( p_n = 1.03 p_{n-1} + 20 \) recursively starting from \( p_0 = 2000 \).

Step 1: Understand the Problem

The pond starts with 2000 trout, and each month 20 trout are added. The population also grows by 3% each month. We need to use the given recursive sequence to find the population after a certain number of months, starting from the initial population.

Step 2: Identify Initial Value

The initial trout population is given by the sequence as \( p_0 = 2000 \). This is our starting point for the population calculations.

Step 3: Understand Recursive Formula

The recursive formula provided is \( p_n = 1.03 p_{n-1} + 20 \). This means that the population in any month \( n \) depends on the previous month's population (\( p_{n-1} \)) multiplied by 1.03 (accounting for the 3% growth) plus the 20 new trouts added each month.

Step 4: Calculate Population for First Month

Using the initial value, calculate the population for the first month \( p_1 = 1.03 \times 2000 + 20 = 2060 \).

Step 5: Calculate Population for Second Month

Using the population from the first month, calculate for the second month \( p_2 = 1.03 \times 2060 + 20 = 2141.8 \).

Step 6: Calculate Population for Third Month

Using the population from the second month, calculate for the third month \( p_3 = 1.03 \times 2141.8 + 20 \approx 2225.054 \).

Step 7: Generalize the Formula

Continue the calculation process in the same manner for subsequent months using the formula \( p_n = 1.03 p_{n-1} + 20 \).

Key Concepts

Recursive Sequences

A recursive sequence is a set of numbers in which each term depends on the previous terms following a consistent rule or formula. In the trout population problem, the number of trouts in the pond each month is calculated based on the number of trouts from the previous month and applying the formula provided. This particular sequence starts with an initial population, then adds a certain growth rate percentage and an additional fixed number of trouts each month. The recursive formula for the trout population is given by:
\( p_0 = 2000 \)
\( p_n = 1.03 p_{n-1} + 20 \)
- **\( p_{n-1} \)** represents the population from the previous month- The term **1.03** accounts for 3% growth- An additional **20** trout is added each month.

Exponential Growth

Exponential growth occurs when the growth rate of a population is proportional to its current size. In the context of the trout population, the major concept of exponential growth is observed through the multiplication of the previous population by a growth factor of 1.03 (which corresponds to a 3% growth rate). This means as the population increases, the amount added due to the growth rate also increases. For example:

• When the initial population is 2000, a 3% increase is 60 trout.
• When the population becomes 2060, the 3% increase is approximately 61.8 trout.

This demonstrates how quickly the population can increase over time, especially when combined with a consistent addition of new population members like the 20 extra trouts added monthly.

Initial Value Problem

An initial value problem involves specifying initial conditions to determine the specific solution to a differential equation or a recursive sequence. For the trout population, the initial value is the starting population of trout, which is given as 2000. This initial value is crucial as it sets the baseline for subsequent calculations. Starting from the initial value, the recursive formula is applied to compute future populations. Here's a concise reasoning step-by-step:

1. Start by identifying the initial population: \( p_0 = 2000 \).
2. Use the recursive formula to find the population for the first month: \( p_1 = 1.03 \times 2000 + 20 = 2060 \).
3. Apply the formula iteratively for each subsequent month using the newly calculated population as the new previous term.

Population Modeling

Population modeling is a mathematical approach to simulate and understand the dynamics of how populations change over time. Using recursive sequences and exponential growth concepts, we can create models for predicting future populations. In this case of trout in a pond, we model their growth using both the natural growth rate and external additions. Essential steps for effective population modeling involve:

• Determining the initial population size.
• Identifying relevant growth rates and additional factors (like hatchery additions).
• Constructing a formula to predict future populations based on previous data.

Using our recursive sequence, the population model in this scenario is built with the formula \( p_n = 1.03 p_{n-1} + 20 \). This allows us to predict how the trout population will evolve monthly, giving valuable insights into managing and sustaining the population growth in the pond.