Problem 45 - College Algebra

Question

In \(2004,\) a school population was \(1,001\) . By 2008 the population had grown to \(1,697\) . Assume the population is changing linearly. a. How much did the population grow between the year 2004 and 2008\(?\) b. How long did it take the population to grow from \(1,001\) students to \(1,697\) students? c. What is the average population growth per year? d. What was the population in the year 2000\(?\) e. Find an equation for the population, \(P,\) of the school \(t\) years after 2000 . f. Using your equation, predict the population of the school in 2011.

Step-by-Step Solution

Answer

The population grew by 696 students in 4 years, at an average of 174 students per year. The population in 2000 was 305, and in 2011, it is predicted to be 2219.

Step 1: Calculate Population Growth (Part a)

To find out how much the population grew between 2004 and 2008, subtract the population in 2004 from the population in 2008. The population in 2008 is 1697 and in 2004 it was 1001.The growth in population is given by:\[ 1697 - 1001 = 696 \]Thus, the population grew by 696 students between 2004 and 2008.

Step 2: Calculate Time Duration (Part b)

To find how long it took for the school population to grow from 1001 to 1697, subtract the year 2004 from the year 2008.\[ 2008 - 2004 = 4 \]It took 4 years for the population to grow from 1001 to 1697 students.

Step 3: Calculate Average Population Growth Per Year (Part c)

To calculate the average population growth per year, divide the total population growth by the number of years it took to grow.The population growth per year is:\[ \frac{696}{4} = 174 \]So, the average population growth per year is 174 students.

Step 4: Determine Population in 2000 (Part d)

Since the population is changing linearly, we can calculate the population in 2000 by working backwards from 2004.The number of years between 2000 and 2004 is 4. The population growth per year is 174.Therefore, the decrease in population from 2004 to 2000 is:\[ 174 \times 4 = 696 \]Subtract this from the population in 2004:\[ 1001 - 696 = 305 \]The population in the year 2000 was 305 students.

Step 5: Find the Linear Equation for Population (Part e)

The population changes linearly, so the equation for population, \( P(t) \), is in the form of a linear equation, \( P(t) = mt + b \).Here, \( m \) represents the growth rate per year, which we calculated as 174. The value of \( b \) is the population at \( t=0 \), which is the year 2000 population of 305.Thus, the equation becomes:\[ P(t) = 174t + 305 \]

Step 6: Predict the Population in 2011 (Part f)

Use the equation \( P(t) = 174t + 305 \) to predict the population in 2011.First, calculate the value of \( t \) for 2011 by subtracting 2000 from 2011:\[ t = 2011 - 2000 = 11 \]Plug \( t = 11 \) into the equation:\[ P(11) = 174 \times 11 + 305 \]Calculate the result:\[ P(11) = 1914 + 305 = 2219 \]So, the predicted population in 2011 is 2219 students.

Key Concepts

Population Growth

Population growth refers to the change in the number of individuals in a population over time. In this context, we are looking at the number of students in a school from one year to another. Between 2004 and 2008, the school's population increased from 1,001 to 1,697. This growth can be easily calculated by subtracting the initial population from the final population.

Understanding population growth is crucial. It provides insight into how resources and planning will need to adjust to handle a growing number of students. Factors such as classroom space, teachers, and materials all depend on accurate growth predictions.

Linear Equations

Linear equations are mathematical expressions that describe a straight line when graphed on a coordinate plane. They generally have the form: \[ y = mx + b \] where \( m \) is the slope (rate of change), \( x \) is the independent variable, and \( b \) is the y-intercept (starting value). In our population growth scenario, the equation becomes:

\[ P(t) = 174t + 305 \]

Here, \( P(t) \) represents the population at time \( t \) years after the year 2000. The slope, \( 174 \), is the average yearly increase, and \( 305 \) is the starting population in 2000. Linear equations are vital in modeling real-life scenarios where there is a constant rate of change.

Average Growth Rate

The average growth rate is a useful metric for determining how quickly a population increases over a specified period. You calculate it by dividing the total growth by the number of years over which the growth occurred.

In our exercise, the total increase in population between 2004 and 2008 was 696 students. This took place over 4 years. Therefore, the average growth rate per year is:

\[ \frac{696}{4} = 174 \]

The rate of 174 students per year is the constant factor in the linear equation, representing how predictably the population grows annually. Understanding the average growth rate helps in planning for future needs.

Predictive Modeling

Predictive modeling uses existing data to forecast future outcomes. When we applied it in our exercise, we used the linear equation:

\[ P(t) = 174t + 305 \]

to predict population in future years. By calculating the population for 2011, we found:

\[ P(11) = 174 \times 11 + 305 = 2219 \]

This equation allowed us to forecast the school’s population based on past trends. Predictive modeling is incredibly useful in education as it helps administrators plan for resources and accommodate growth. It can guide decisions on whether to expand facilities or hire additional staff.