Problem 61 - College Algebra with Corequisite Support

Question

The fox population in a certain region has an annual growth rate of \(9 \%\) per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year \(2020 ?\)

Step-by-Step Solution

Answer

The predicted fox population in 2020 is 47,621.

Step 1: Understand the growth formula

The population growth can be modeled using the exponential growth formula, which is given by \( P = P_0 (1 + r)^t \) where \( P \) is the population after time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the number of years.

Step 2: Identify initial values

From the problem, we know that the initial population, \( P_0 \), is 23,900 foxes, the growth rate \( r \) is 9\% (or 0.09 as a decimal), and we need to find the population in 2020, starting from 2012, so \( t = 2020 - 2012 = 8 \) years.

Step 3: Substitute the values into the formula

Substitute the values into the formula: \( P = 23900 \times (1 + 0.09)^8 \).

Step 4: Calculate the expression inside the parentheses

Compute \( 1 + 0.09 = 1.09 \).

Step 5: Raise to the power of \( t \)

Raise 1.09 to the power of 8: \( 1.09^8 \approx 1.9926 \).

Step 6: Multiply by the initial population

Multiply the initial population by this result: \( 23900 \times 1.9926 \approx 47621.34 \).

Step 7: Round to the nearest whole number

Since population is typically a whole number, round 47621.34 to 47,621.

Key Concepts

Population Growth

Population growth is a fascinating natural phenomenon that describes how the number of individuals in a species population increases over time. In biology and ecology, this growth is often due to factors such as birth rates, immigration, and resource availability.
In general, populations tend to grow in environments where there is plenty of food and few predators.

Understanding population growth is crucial for managing wildlife, planning conservation efforts, and ensuring sustainable use of natural resources.

• It helps estimate how a population might change over time.
• It provides insights into the health and dynamics of ecosystems.
• It serves a role in predicting potential overpopulation issues.

In mathematical terms, population growth can be linear or exponential, where exponential growth is often observed in natural populations like the foxes in the given exercise.

Growth Rate

The growth rate is a key component in understanding population dynamics. It is defined as the rate at which a population increases in size during a specified time period. Growth rates are typically expressed as a percentage, indicating how much the population grows per year.

In terms of exponential growth, the growth rate is used to represent the consistent percentage increase over time. For example, a 9% growth rate means that the population increases by 9% per year relative to its current size.

Grasping the concept of growth rate allows us to make predictions about future population sizes and understand the factors influencing growth. It can provide insights into whether a population might:

• Grow rapidly.
• Stay stable.
• Decline.

In the case of the fox population, knowing the growth rate of 9% allows us to calculate how much the population will increase each year.

Exponential Growth Formula

The exponential growth formula is a mathematical equation used to predict future values in scenarios where growth is proportional to the current amount. It is expressed as \( P = P_0 (1 + r)^t \), where:

• \( P \) is the future population size.
• \( P_0 \) is the starting population size.
• \( r \) is the annual growth rate as a decimal.
• \( t \) is the time in years over which growth occurs.

This formula captures the essence of exponential growth, showing how small consistent increases can lead to substantial changes over time.

In the fox population problem, we used this formula to find the population after 8 years. By substituting in known values, such as an initial population of 23,900 and a growth rate of 0.09, and solving across 8 years, the calculation grants us a prediction of the number of foxes in 2020. This demonstrates the powerful predictive capability of the exponential growth formula.