Problem 46 - College Algebra with Corequisite Support

Question

In \(2003,\) a town's population was 1431. By 2007 the population had grown to 2134 . Assume the population is changing linearly. (a) How much did the population grow between the year 2003 and \(2007 ?\) (b) How long did it take the population to grow from 1431 people to 2134 people? 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 town \(t\) years after 2000 . (f) Using your equation, predict the population of the town in 2014 .

Step-by-Step Solution

Answer

a) 703 people; b) 4 years; c) 175.75 people/year; d) 904 people; e) \(P(t) = 175.75t + 904\); f) Approximately 3361 people.

Step 1: Calculate Population Growth

To find out how much the population grew between 2003 and 2007, subtract the population in 2003 from the population in 2007.\[\text{Population Growth} = 2134 - 1431 = 703\]

Step 2: Determine Time Period

Calculate the time period over which the population grew by subtracting the initial year from the final year.\[\text{Time Period} = 2007 - 2003 = 4 \text{ years}\]

Step 3: Calculate Average Population Growth per Year

The average population growth per year is the total population change divided by the number of years.\[\text{Average Growth} = \frac{703}{4} = 175.75 \text{ people per year}\]

Step 4: Calculate Population in 2000

To find the population in 2000, use the average growth rate to work backwards from 2003. From 2003 to 2000 is 3 years.\[\text{Population in 2000} = 1431 - (175.75 \times 3) \approx 903.75\] Since the population must be an integer, we round this to 904.

Step 5: Derive the Population Equation

Using the information about growth from 2000, formulate a linear equation. The general form is \(P(t) = mt + b\), where \(m\) is the rate of change (175.75 people per year), and \(b\) is the initial population (in 2000). Therefore, the equation becomes:\[P(t) = 175.75t + 904\]

Step 6: Predict Population in 2014 Using the Equation

Substitute \(t = 14\) (since 2014 is 14 years after 2000) into the equation \(P(t) = 175.75t + 904\).\[P(14) = 175.75 \times 14 + 904 = 3360.5\] The population is approximately 3361, rounding to the nearest whole number.

Key Concepts

Population Growth Calculation

Population growth calculation is a fundamental step in understanding how a population changes over time. In our example, we begin by identifying the change in population between two given years, 2003 and 2007. To do this, we simply subtract the earlier population from the later population. This gives us a net increase, which represents how much the population has grown over the period.

For instance, the population in 2003 was 1431, and by 2007 it had increased to 2134. The calculation would therefore be:

• Subtract the population of 2003 from that of 2007: 2134 - 1431.
• This results in a growth of 703 people over the 4-year span.

Understanding this basic subtraction helps us see the total growth, laying the groundwork for further analysis, such as determining the yearly average growth.

Average Growth Rate

Determining the average growth rate is crucial for understanding the pace at which the population increases each year. Once we've calculated the total growth over a specific period, we can then calculate the average growth per year by dividing the total growth by the number of years over which this growth occurred.

Consider our example, where the population increased by 703 people from 2003 to 2007, spanning 4 years:

• Total growth is 703 people.
• Time period is 4 years.

This mean, the calculation is:

• Average growth = \( \frac{703}{4} = 175.75 \) people per year.

Finding the average growth rate aids in making predictions about future population trends, as it simplifies the rate of change to a constant yearly value.

Linear Equation Formulation

Creating a linear equation that represents population growth is a powerful tool in predictive modeling. A linear equation expresses how the population changes over time with a steady rate of growth. Such an equation is written in the form:

• \( P(t) = mt + b \)

Here, \( m \) is the average yearly increase (or slope), and \( b \) is the starting population (the y-intercept).

In our case, we used the information that the initial population in 2000 was approximately 904 people, and the average yearly increase was 175.75 people. Substituting these values in, we have:

• \( P(t) = 175.75t + 904 \)

This linear model serves as a mathematical representation of the town's population over time, allowing us to further explore and predict future trends.

Predictive Modeling in Algebra

With a linear equation established, predictive modeling becomes straightforward. By substituting different values of \( t \) (years since 2000) into the equation, we can estimate the population for future or intermediate years.

For example, to find the projected population in 2014, calculate as follows:

• Substitute \( t = 14 \) into the equation \( P(t) = 175.75t + 904 \).
• Solve for \( P(14) \):
• \( P(14) = 175.75 \times 14 + 904 = 3360.5 \)

Rounding, the town's estimated population in 2014 is 3361 people. Predictive modeling gives us insight into future population challenges and aids in urban planning by simulating possible future scenarios.