Problem 18 - Algebra A Combined Function

Question

Solve. Unless noted otherwise, round answers to the nearest whole. The size of the rat population of a wharf area grows at a rate of \(8 \%\) monthly. If there are 200 rats in January, find how many rats should be expected by next January.

Step-by-Step Solution

Answer

Approximately 504 rats.

Step 1: Understand the Problem

We are given the initial size of the rat population as 200 in January and that it grows at a rate of 8% each month. We are asked to find the population size 12 months later, in the next January.

Step 2: Identify the Formula for Exponential Growth

The population grows by a fixed percentage monthly, which can be modeled by the exponential growth formula: \[ P = P_0 (1 + r)^t \] where \( P \) is the future population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in months.

Step 3: Substitute Given Values into the Formula

Given: \( P_0 = 200 \) (starting population), \( r = 0.08 \) (8% growth rate), \( t = 12 \) months.Substitute these into the formula:\[ P = 200 (1 + 0.08)^{12} \]

Step 4: Calculate the Exponential Factor

First, calculate \( (1 + 0.08)^{12} \):\[ (1.08)^{12} \approx 2.518 \] This factor represents the total growth over 12 months.

Step 5: Compute the Final Population

Multiply the initial population by the calculated exponential factor:\[ P = 200 \times 2.518 \approx 503.6 \]Round this to the nearest whole number, resulting in a final population estimate of approximately 504 rats.

Key Concepts

population modeling

Population modeling is a powerful tool used to understand how populations change over time. It involves using mathematical formulas and data to make predictions about the future size of a population. In the context of the rat population in a wharf area, we used an exponential growth model to project future population numbers.

• The purpose of population modeling is to help predict future dynamics.
• It can inform management and control strategies.
• By knowing the current population and growth rate, we can anticipate future changes.

For example, by applying the formula for exponential growth, we could estimate how many rats would be in the area in one year, if the rate of growth remains constant. This kind of modeling is crucial for effective decision-making in wildlife management and urban planning.

growth rate

The growth rate is a key element in population modeling. It signifies the rate at which a population increases over a set period. In this exercise, the growth rate was 8% per month, meaning each month the rat population increased by 8% of its size from the previous month.

• Expressed as a decimal (8% = 0.08), it is used in calculations as a multiplier.
• An 8% monthly growth rate suggests relatively rapid increases over time.
• The growth rate can significantly affect predictions made in population models.

Consistent growth rates like the one in our problem can be seen in natural populations assuming unlimited resources and no limiting factors affecting growth. Understanding and using the growth rate properly allows us to calculate future population sizes accurately using exponential growth formulas.

exponential factor

In the problem, the exponential factor plays a vital role in determining how the population will look after a given period. It arises from the exponential growth formula where the base \[ (1 + r) \] is raised to the power of the time interval \( t \). In our scenario, the exponential factor for 12 months was calculated as:\[ (1.08)^{12} \approx 2.518 \].

• This factor tells us how many times larger the population will be after a given period.
• It includes both the original population size as well as accumulated growth.
• The total population estimate is obtained by multiplying the initial population by this factor.

The concept of an exponential factor is essential because it encapsulates the cumulative effect of consistent percentage increases over time, making it easier to see the overall growth at the end of the specified period, such as the one year in our rat population example.