Problem 61 - College Algebra

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 approximately 47,610.

Step 1: Understand Annual Growth Rate Formula

To calculate future populations given a constant growth rate, use the formula for exponential growth:\[ P = P_0 (1 + r)^t \]where \(P\) is the future population, \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the number of years.

Step 2: Identify Known Values

From the problem, we know the following:- Initial population \(P_0 = 23,900\)- Growth rate \(r = 0.09\) (9% expressed as a decimal)- Years \(t = 2020 - 2012 = 8\)

Step 3: Insert Values into the Formula

Substitute the known values into the exponential growth formula:\[ P = 23,900 \times (1 + 0.09)^8 \]

Step 4: Calculate the Future Population

First, calculate \((1 + 0.09)^8\):\[ (1 + 0.09)^8 = 1.09^8 \approx 1.992563 \]Then, multiply by the initial population to find \(P\):\[ P = 23,900 \times 1.992563 \approx 47,610 \]

Step 5: Provide the Final Answer

Therefore, the predicted fox population in the year 2020 is approximately 47,610.

Key Concepts

Annual Growth Rate

Understanding the annual growth rate is key when predicting how populations change over time. The annual growth rate is expressed as a percentage and tells us how much a population increases each year. When a population grows by a certain percentage, it becomes larger in the following year, and subsequent growth is based on this larger size.
The formula commonly used to calculate such growth is:

• \( P = P_0 (1 + r)^t \)

Where:

• \(P\) is the future population we want to find.
• \(P_0\) is the initial population at the start of our observation.
• \(r\) is the growth rate, represented as a decimal (so 9% becomes 0.09).
• \(t\) is the time in years over which growth occurs.

This formula is derived from the concept of exponential growth, where populations multiply by the growth factor \((1 + r)\) each year. This is different from linear growth, as it accounts for compounding growth.

Fox Population Prediction

Predicting the fox population involves understanding and using the annual growth rate effectively. Suppose we are tasked with predicting the fox population in a particular year, taking the initial population and expected growth rate into account.
In our case, the initial population of foxes in 2012 is known to be 23,900, and the annual growth rate is 9%. We must predict the population by a future year, which is 2020 in this scenario.
Using the formula \( P = P_0 (1 + r)^t \), we identify the known values:

• \(P_0 = 23,900\)
• \(r = 0.09\) (since 9% growth rate)
• \(t = 2020 - 2012 = 8\) years

Substituting these into the formula allows us to calculate the future population, taking into account how each year's population increase compounds on the last.

Future Population Calculation

Calculating the future population using exponential growth can seem tricky, but it's straightforward once you break it down. Let's walk through the steps to find out how many foxes there might be in 2020.
First, calculate the yearly multiplier, \( (1 + 0.09) = 1.09 \). Each year, the population will be 1.09 times the size it was the previous year. Next, because we are looking 8 years into the future, calculate:

• \(1.09^8 \approx 1.992563\)

This tells us that over 8 years, the population nearly doubles due to the combined effect of the annual growth rate.
Finally, to find the population in 2020, multiply the initial population by this factor:

• \( P = 23,900 \times 1.992563 = 47,610 \)

Thus, the predicted population is approximately 47,610 foxes in 2020. Each stage of the calculation considers how growth accumulates over each year, providing a precise estimate of future populations.