Problem 24 - Algebra 1

Question

A population of 30 mice is released in a wildlife region. The population doubles each year for 4 years. What is the population after 4 years?

Step-by-Step Solution

Answer

The population of mice after 4 years is 480.

Step 1: Understand the situation

The population starts at 30 mice and doubles every year, which is a basic example of exponential growth. Given is that this happens over a period of 4 years.

Step 2: Apply the exponential growth formula

In general, the formula for exponential growth can be defined as the initial value times growth rate to the power of the duration, this can be written as: \(N = N0 * 2^t\). Where \(N\) is the final population, \(N0\) is the initial population, and \(t\) is the time, which is 4 years in our case.

Step 3: Substitute the given values and solve

Substitute the given values \(N0 = 30\) and \(t = 4\) into the equation. \(N = 30 * 2^4\). Compute \(2^4 = 16\), then multiply \(30 * 16 = 480\).

Key Concepts

Population Growth

Population growth is a key concept in studying how populations of living organisms change over time. It can refer to any type of organism, from bacteria to humans, but here we talk about mice. When we mention growth, we often refer to how the number of individuals in a population increases.

• Starting with a certain number of individuals, such as 30 mice, we observe how this number changes annually.
• In this scenario, we see a doubling trend, which means that each year the population becomes two times larger.

The population doubling each year highlights an important growth trend. This type of growth is common in many natural settings, especially where limited resources or predators control the population. Not all growth can continue indefinitely the same way due to environmental constraints, but for initial calculations, like our example, it's purely mathematical.

Exponential Functions

Exponential functions are a crucial tool in mathematics, especially for modeling situations where growth follows a constant multiplicative rate. This trait makes them perfect for describing how our mouse population behaves.

• In mathematical terms, exponential growth happens when the increase rate of a value is proportional to the current value.
• For our mice, this means each year, the population grows based on its current size.

The exponential growth formula is expressed as:\[N = N_0 \times 2^t\]Where:

• \(N\) is the population after time \(t\),
• \(N_0\) is the starting number of mice,
• and \(t\) is the number of years.

In our mouse scenario, when the population doubles annually, we use the base of the exponent 2. This reflects the constant doubling each year.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world situations. This approach helps us understand complex systems through simple equations and formulas.

• We begin by understanding the real-world phenomenon, like our growing mouse population.
• Then, we translate this into a mathematical form, such as an exponential function.

In mathematical modeling, accuracy depends on how well the model reflects reality. For the mouse population example, our simple model uses the formula:\[N = 30 \times 2^4\]

• Here, \(30\) is the initial population, \(2\) is the growth factor (doubling), and \(4\) is the duration in years.
• This model predicts a population of 480 mice after 4 years, illustrating the power of exponential models to project future outcomes.

Through modeling, we gain insights into population dynamics, exploring not just what will happen, but understanding the rate and impact of change over time.