Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

The populations \(P\) (in thousands) of Pineville, North Carolina, from 2006 through 2012 can be modeled by \(P=5.4 e^{k t},\) where \(t\) is the year, with \(t=6\) corresponding to \(2006 .\) In \(2008,\) the population was 7000. (Source: U.S. Census Bureau) (a) Find the value of \(k\) for the model. Round your result to four decimal places. (b) Use your model to predict the population in 2018 .

Step-by-Step Solution

Answer

The value of 'k' for the model is approximately 0.2198. The predicted population of Pineville in 2018, using the model, is approximately 20876 thousand or 20.9 million.

Step 1: Establish Population Data for 't' and 'P' (Year and Population)

From the problem, we know the population of Pineville in 2006 (where \(t=6\)) was 5400 (since 'P' is in thousands). We're also given that the population was 7000 in 2008. Hence, we can translate 2008 to \(t=8\).

Step 2: Find the Constant 'k'

We know that 'P' in 2008 was 7000. So, we substitute \(P=7000\), \(t=8\) in the given model equation \(P = 5.4 \cdot e^{k t}\). We need to solve this equation for 'k'. Solving this gives \(k = \ln( \frac {7000}{5.4})/8\) or approximately \(k = 0.2198\).

Step 3: Predicting Population in 2018

With our found 'k' value from step 2, we have our exponential model as \(P = 5.4 \cdot e^{0.2198 t}\). To predict the population in 2018, \(t = 18\). Substituting this 't' into our model gives \(P = 5.4 \cdot e^{0.2198 \cdot 18}\), which is approximately 20876. This means the predicted population in 2018 is around 20876 thousand or approximately 20.9 million.

Key Concepts

Population Modeling

Population modeling is a fascinating way to understand how a population changes over time. It helps us predict future shifts based on current data and trends.
This is particularly useful for city planning, resource allocation, and understanding the dynamics of growth.

In our example, we have the town of Pineville, North Carolina. We know the population at two specific years: 2006 and 2008.
Using this information, we express the population as a function of time, denoted as \(P = 5.4 e^{kt}\).

• The constant term 5.4 represents the initial population (in thousands) when \(t=6\), or in 2006.
• The variable \(t\) represents time in years since 2006, so it is a measure of how long the population has been growing.

With these parameters, the population model allows us to estimate the population at any point in time, which is invaluable for future planning and development.

Exponential Functions

Exponential functions are central to many real-world applications, including population models.
They help describe growth processes that increase at a consistent rate over time.

In our case, the exponential function is expressed as \(P = 5.4 e^{kt}\). Here's how it works:

• The term \(e^{kt}\) represents exponential growth. 'e' is a mathematical constant approximately equal to 2.71828.
• The exponent \(kt\) dictates the rate and duration of growth, where 'k' is the growth constant we need to determine.

In the population example, the presence of the exponential term indicates that the population grows by a constant percentage each year.
This is why exponential models are so powerful—they highlight how small, consistent changes can lead to large outcomes over time.
Understanding exponential functions lets us better predict and prepare for such changes.

Predictive Modeling

Predictive modeling involves using past data to make informed guesses about the future.
It is a fundamental tool in many industries, from economics to weather forecasting.

Here, we apply predictive modeling to estimate Pineville's population in 2018.
Once we have determined the growth factor \(k\), we can project the population a decade into the future.
This involves plugging the value of \(t\) representing the year 2018 (\(t=18\)) into our model: \(P = 5.4 e^{0.2198 \times 18}\).

With predictive modeling:

• We forecast the population will grow significantly, reaching nearly 20.9 million by 2018.
• This projection helps town planners in decision making, ensuring they prepare for increased population needs.

In summary, predictive modeling transforms historical trends into actionable insights, guiding future decisions and policies.