Question

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. The state of Texas had the largest increase in population of any state in the U.S. from 2000 to 2010. During that decade, Texas grew from 20.9 million in 2000 to 25.1 million in \(2010 .\) Use an exponential growth model to predict the population of Texas in 2025 .

Short answer

Answer: The predicted population of Texas in 2025 is approximately 31.43 million people.

Step-by-step solution

Identifying the reference point and units of time

The reference point \((t=0)\) is the starting point of our analysis. In this case, it is the year 2000 when the population was 20.9 million. The units of time in this exercise are years.

Create a general exponential growth function

An exponential growth function has the form: \(P(t) = P_0 * e^{k*t}\) Where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population (at \(t=0\)), \(e\) is the base of natural logarithms (approximately \(2.718\)), \(k\) is the growth rate, and \(t\) is the time in years.

Find the parameters using the given data

We are given that the population in 2000 (when \(t=0\)) was 20.9 million and in 2010 (when \(t=10\)) was 25.1 million. We can use these values to find parameter \(k\). Using the formula and the given data for 2010 (assuming \(t=10\)): \(25.1 = 20.9 * e^{k*10}\) Now we need to solve for \(k\). First, divide both sides by \(20.9\) to isolate the exponential term: \(1.20095 = e^{10k}\) Now, take the natural logarithm of both sides: \(\ln(1.20095) = 10k \implies k = \frac{\ln(1.20095)}{10} \approx 0.0183\)

Write the exponential growth function with the determined parameters

Now that we have our parameter \(k\), we can write our exponential growth function as: \(P(t) = 20.9 * e^{0.0183 * t}\)

Predict the population in 2025

To predict the population in 2025, we need to plug in \(t=25\) into our function: \(P(25) = 20.9 * e^{0.0183 * 25} \approx 31.43\) So, the population of Texas in 2025 is predicted to be approximately 31.43 million people.

Key concepts

Population Modeling

Population modeling is a fundamental tool in understanding how populations change over time. It helps demographers, scientists, and policymakers make predictions about the future based on current and historical data. In the given exercise, population modeling involves understanding the population increase in Texas from 2000 to 2010 and making future predictions.
The process starts by collecting data about the population at different times. For Texas, the data were recorded for the years 2000 and 2010, showing a growth from 20.9 million to 25.1 million.
Next, a mathematical model is chosen to describe this growth. In this case, an exponential growth model is used because populations often grow exponentially under ideal conditions.
Such models assume that the rate of population growth is proportional to the size of the population. This leads to the formulation of an exponential function that can predict future population sizes based on past growth trends.

Growth Rate Calculation

Calculating the growth rate is essential for constructing an accurate population model. The growth rate indicates how quickly a population is increasing over time, expressed as the constant "k" in exponential functions.
To find the growth rate, we use the known population figures and their corresponding times. From the exercise, we know the population in Texas was 20.9 million in 2000 and rose to 25.1 million in 2010.
The growth rate is calculated as follows:

• Start with the exponential growth formula: \(P(t) = P_0 * e^{k*t}\).
• In 2010, rearrange the formula to solve for \(k\) using the known populations: \(25.1 = 20.9 * e^{10k}\).
• By dividing both sides by 20.9, and taking the natural logarithm, you isolate \(k\): \(\ln(1.20095) = 10k\). Solving gives \(k \approx 0.0183\).

The growth rate \(k\) can now be used to predict future populations and is a measure of how fast the population will likely to continue to grow.

Exponential Functions

Exponential functions are mathematical expressions widely used in population modeling and many other fields like finance and environmental science.
In the context of population growth, an exponential function can predict how a population changes over time. The exponential growth formula used in the exercise is:\[P(t) = P_0 * e^{k*t}\]Here:

• \(P(t)\) is the population at time \(t\).
• \(P_0\) is the initial population at the starting time (e.g., year 2000).
• \(e\) is the base of natural logarithms, approximately equal to 2.718.
• \(k\) is the growth rate.
• \(t\) is the time elapsed since the starting point.

In the exercise, this function is used to predict Texas' population in 2025. By inserting \(t=25\), the time since the year 2000, into the function, you can estimate the population for that year. The exponential function provides a sophisticated way to handle varying rates of growth, as it assumes continuous, proportional changes over time rather than linear growth. This makes it particularly useful for modeling real-world phenomena like population changes.