Problem 48 - Precalculus Mathematics for Calculus

Question

Mouse Population A certain breed of mouse was introduced onto a small island with an initial population of 320 mice, and scientists estimate that the mouse population is doubling every year. (a) Find a function that models the number of mice after \(t\) years. (b) Estimate the mouse population after 8 years.

Step-by-Step Solution

Answer

The function is \( P(t) = 320 \cdot 2^t \). The population after 8 years is 81,920 mice.

Step 1: Understand the problem

We need to identify a function that can calculate the mouse population when it doubles every year, starting with an initial population of 320 mice. Additionally, we need to estimate the population after 8 years using this function.

Step 2: Identify the type of function

Since the population doubles every year, the problem can be modeled with an exponential function of the form \( P(t) = P_0 \cdot 2^t \), where \( P_0 \) represents the initial population, and \( t \) represents the number of years.

Step 3: Find the function

Substitute the initial population into the function: \( P_0 = 320 \). Thus, the function becomes \( P(t) = 320 \cdot 2^t \). This function will model the mouse population over time.

Step 4: Calculate the population after 8 years

Substitute \( t = 8 \) into the function: \( P(8) = 320 \cdot 2^8 \). Simplify this expression to calculate the population.

Step 5: Perform the calculation

Calculate \( 2^8 = 256 \). So \( P(8) = 320 \cdot 256 = 81920 \). Thus, the mouse population after 8 years is 81,920 mice.

Key Concepts

Mathematical Modeling

Mathematical modeling is a process that involves creating mathematical formulas and structures to represent real-world scenarios. This helps us analyze and make predictions about these situations. In the context of the mouse population problem, we're using mathematical modeling to understand how the number of mice changes over time.

When setting up a mathematical model, it's crucial to identify the key factors affecting the system. For our mice, the primary factors include the initial population and the rate at which the population increases (doubling each year).

The goal of the model is to create a function that accurately describes the dynamics of the system. This involves using known mathematical methods, like exponential functions, as they can represent growth processes effectively. With a sound mathematical model, predictions such as future population size are possible by simply plugging in the desired number of years into the function.

Population Growth

Population growth refers to the increase in the number of individuals in a population, and it can be influenced by various factors.

• Natural birth and death rates
• Migration to and from the area
• Environmental conditions

In the given scenario, we look at an idealized case where the mouse population grows exponentially. Every year, the population doubles, showcasing an unrestrained growth pattern mainly determined by the doubling rate.

Such a growth model assumes no limits on resources, allowing the population to expand exponentially. However, in real-world situations, factors like resources, space limitations, or disease often slow down growth as the population reaches the carrying capacity of the environment.

By understanding these dynamics, one can apply such population growth models in ecology to estimate how certain species might grow or decline in specific habitats.

Exponential Functions

Exponential functions are a class of mathematical functions used to model situations where quantities grow or decay at a constant relative rate over time. They have the general form:

• \( f(t) = a \cdot b^t \)

Here, \( a \) is the initial quantity, \( b \) is the growth (or decay) factor, and \( t \) represents time.

In our mouse population example, the function is \( P(t) = 320 \cdot 2^t \). Here, \( 320 \) is the starting population, and \( 2 \) is the base that indicates the population doubles each year.

Exponential functions are powerful because they can predict long-term behaviors, such as the projected population size after a certain number of years. Their distinctive feature is the rapid speed at which quantities grow or decline; in educational terms, it's like a snowball effect where changes become bigger and bigger over time.

Understanding exponential functions equips students with the tools to model various real-world phenomena, from financial growth predictions to human population studies.