Problem 74 - Algebra and Trigonometry

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

6013 catfish after 12 months.

Step 1: Understand the Initial Conditions

We start with 5000 catfish in the pond, which is represented as \(P_0 = 5000\). This forms the basis of our recursive formula for the population.

Step 2: Formulate the Monthly Growth

The number of catfish increases by 8% each month. If the population at month \(n-1\) is \(P_{n-1}\), then the population before harvesting for month \(n\) is \(1.08 \times P_{n-1}\). This is because an 8% increase is mathematically represented by multiplying by 1.08.

Step 3: Account for Monthly Harvest

Each month, 300 catfish are harvested from the pond. Therefore, after growth is calculated, we subtract 300 catfish to give the population at month \(n\). Thus, the recursive formula becomes \(P_n = 1.08P_{n-1} - 300\).

Step 4: Calculate Population after 12 Months

Starting with \(P_0 = 5000\), use the recursive formula \(P_n = 1.08P_{n-1} - 300\) to calculate \(P_1\), \(P_2\), and so on, up to \(P_{12}\).

Step 5: Calculate Sequentially

Compute iteratively for each month:- \(P_1 = 1.08 \times 5000 - 300 = 5100 - 300 = 4800\)- \(P_2 = 1.08 \times 4800 - 300 = 5184 - 300 = 4884\)Continue this process...Finally, - \(P_{12} = 1.08 \times 5845 - 300 = 6312.60 - 300 = 6012.60\)Round to \(6013\) since the population must be a whole number.

Key Concepts

Population Growth

Population growth in ecological settings refers to the change in the number of individuals in a species population over time. Understanding this can help manage resources, like when a fish farmer tracks catfish numbers in a pond. In this exercise, the catfish population grows each month.

• Starting population, known as the initial population: 5000 catfish.
• Growth is affected by reproduction and environmental factors.

To illustrate population growth, we use a recursive method that layers the effect of each month's growth and harvesting. This helps in predicting how many fish remain or increase over varied periods.

Exponential Growth

Exponential growth occurs when the increase in population size becomes more rapid in proportion to the growing total number of individuals. For the fish in the pond, this growth is described as a monthly 8% increase. The growth can be predicted mathematically:

• Population at the start of a month is multiplied by a growth factor.
• In this case, 8% growth means multiplying by 1.08.

As per the recursive formula, each month's population is calculated, maintaining a continuous chain of growth. However, bear in mind that real-life conditions like resources or space would limit infinite growth.

Monthly Harvesting

Monthly harvesting involves regularly removing a set number of individuals from a population. In our example, 300 catfish are harvested each month. This process modifies the natural exponential growth: - After the growth from an 8% increase, 300 fish are subtracted. - This keeps the pond balanced and prevents overpopulation, but still needs precise planning. Regular harvesting can help sustain finite resources but must account for growth rates to ensure the population remains viable.