Select The Correct Answer.Due To An Increasing Population, Developed Land In A Region Is Increasing At A Rate Of $12\%$ Per Year. If There Are Currently 8,500 Square Miles Of Developed Land, Which Equation Models The Square Mileage Of
Introduction
As the world's population continues to grow, so does the demand for developed land. In this scenario, we are given that the developed land in a region is increasing at a rate of 12% per year. With an initial amount of 8,500 square miles of developed land, we need to determine the equation that models the square mileage of developed land over time.
Understanding the Problem
To model the growth of developed land, we can use the concept of exponential growth. Exponential growth occurs when a quantity increases by a fixed percentage over a fixed period of time. In this case, the developed land is increasing by 12% per year.
The Equation for Exponential Growth
The equation for exponential growth is given by:
A(t) = A0 * (1 + r)^t
Where:
- A(t) is the amount of developed land at time t
- A0 is the initial amount of developed land (8,500 square miles)
- r is the rate of growth (12% or 0.12 as a decimal)
- t is the time in years
Substituting the Given Values
Now that we have the equation for exponential growth, we can substitute the given values to get:
A(t) = 8,500 * (1 + 0.12)^t
Simplifying the Equation
To simplify the equation, we can rewrite it as:
A(t) = 8,500 * (1.12)^t
Understanding the Equation
The equation A(t) = 8,500 * (1.12)^t models the square mileage of developed land over time. The (1.12)^t term represents the exponential growth of developed land, while the 8,500 term represents the initial amount of developed land.
Interpreting the Results
To interpret the results, we can plug in different values of t to see how the amount of developed land changes over time. For example, if we plug in t = 1, we get:
A(1) = 8,500 * (1.12)^1 A(1) = 9,440
This means that after one year, the amount of developed land will be approximately 9,440 square miles.
Conclusion
In conclusion, the equation A(t) = 8,500 * (1.12)^t models the square mileage of developed land over time. By understanding the equation and its components, we can interpret the results and make predictions about the growth of developed land in the region.
Example Use Cases
- Predicting Future Growth: Using the equation, we can predict the amount of developed land in the region for future years.
- Comparing Growth Rates: We can compare the growth rate of developed land in this region to other regions to see which one is growing faster.
- Understanding the Impact of Growth: We can use the equation to understand the impact of growth on the environment, economy, and society.
Common Mistakes to Avoid
- Incorrectly Substituting Values: Make sure to substitute the correct values into the equation.
- Not Simplifying the Equation: Simplify the equation to make it easier to understand and work with.
- Not Interpreting the Results: Take the time to interpret the results and make predictions about the growth of developed land.
Additional Resources
- Exponential Growth Calculator: Use an online calculator to calculate the exponential growth of developed land.
- Mathematical Models: Learn more about mathematical models and how they are used to predict and understand complex phenomena.
- Environmental Impact: Learn more about the environmental impact of growth and development.
Frequently Asked Questions: Modeling Population Growth ===========================================================
Q: What is exponential growth?
A: Exponential growth is a type of growth where a quantity increases by a fixed percentage over a fixed period of time. In the context of this problem, the developed land is increasing by 12% per year.
Q: What is the equation for exponential growth?
A: The equation for exponential growth is given by:
A(t) = A0 * (1 + r)^t
Where:
- A(t) is the amount of developed land at time t
- A0 is the initial amount of developed land (8,500 square miles)
- r is the rate of growth (12% or 0.12 as a decimal)
- t is the time in years
Q: What is the significance of the (1 + r)^t term?
A: The (1 + r)^t term represents the exponential growth of developed land. This term takes into account the fixed percentage increase in developed land over time.
Q: How do I interpret the results of the equation?
A: To interpret the results, you can plug in different values of t to see how the amount of developed land changes over time. For example, if you plug in t = 1, you get:
A(1) = 8,500 * (1.12)^1 A(1) = 9,440
This means that after one year, the amount of developed land will be approximately 9,440 square miles.
Q: Can I use this equation to predict future growth?
A: Yes, you can use this equation to predict future growth. Simply plug in the desired value of t and calculate the corresponding amount of developed land.
Q: What are some common mistakes to avoid when using this equation?
A: Some common mistakes to avoid include:
- Incorrectly substituting values into the equation
- Not simplifying the equation
- Not interpreting the results
Q: What are some additional resources that can help me learn more about this topic?
A: Some additional resources that can help you learn more about this topic include:
- Exponential growth calculators
- Mathematical models
- Environmental impact studies
Q: Can I use this equation to compare the growth rates of different regions?
A: Yes, you can use this equation to compare the growth rates of different regions. Simply plug in the desired values of A0, r, and t for each region and calculate the corresponding amount of developed land.
Q: What are some real-world applications of this equation?
A: Some real-world applications of this equation include:
- Predicting population growth
- Understanding the impact of growth on the environment
- Making informed decisions about urban planning and development
Q: Can I use this equation to model other types of growth?
A: Yes, you can use this equation to model other types of growth, such as population growth or economic growth. Simply adjust the equation to fit the specific type of growth you are modeling.