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

Modeling Population Growth: Understanding the Equation for Developed Land

The increasing population of a region can have a significant impact on the development of land. In this scenario, we are given that the developed land in a region is increasing at a rate of $12\%$ per year. We are also told that there are currently 8,500 square miles of developed land. The question is, which equation models the square mileage of developed land over time?

To model the growth of developed land, we need to consider 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:

$$y = ab^x$$

where:

• $y$ is the final value (in this case, the square mileage of developed land)
• $a$ is the initial value (in this case, the current square mileage of developed land, which is 8,500)
• $b$ is the growth factor (in this case, $1 + 0.12 = 1.12$, since the land is increasing by $12\%$ per year)
• $x$ is the time period (in this case, the number of years)

Substituting the Values

Now that we have the equation for exponential growth, we can substitute the values given in the problem:

$$y = 8,500(1.12)^x$$

This equation models the square mileage of developed land over time.

Interpreting the Equation

The equation $y = 8,500(1.12)^x$ tells us that the square mileage of developed land will increase by $12\%$ per year. This means that if we multiply the current square mileage of developed land (8,500) by the growth factor (1.12), we will get the square mileage of developed land after one year.

For example, if we want to find the square mileage of developed land after 5 years, we can substitute $x = 5$ into the equation:

$$y = 8,500(1.12)^5$$

Simplifying the equation, we get:

$$y = 8,500(1.7628)$$

$$y = 14,943.6$$

Therefore, the square mileage of developed land after 5 years will be approximately 14,943.6 square miles.

In conclusion, the equation $y = 8,500(1.12)^x$ models the square mileage of developed land over time. This equation takes into account the increasing population of the region and the resulting growth of developed land. By substituting the values given in the problem, we can use this equation to predict the square mileage of developed land after a certain number of years.

• The equation for exponential growth is $y = ab^x$.
• The growth factor is $1 + 0.12 = 1.12$.
• The equation $y = 8,500(1.12)^x$ models the square mileage of developed land over time.
• The equation can be used to predict the square mileage of developed land after a certain number of years.

The final answer is: $y = 8,500(1.12)^x$
Frequently Asked Questions: Modeling Population 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 the problem, the developed land is increasing by $12\%$ per year, which is an example of exponential growth.

A: The equation for exponential growth is given by:

$$y = ab^x$$

where:

• $y$ is the final value (in this case, the square mileage of developed land)
• $a$ is the initial value (in this case, the current square mileage of developed land, which is 8,500)
• $b$ is the growth factor (in this case, $1 + 0.12 = 1.12$, since the land is increasing by $12\%$ per year)
• $x$ is the time period (in this case, the number of years)

A: To use the equation to predict the square mileage of developed land after a certain number of years, you need to substitute the values into the equation. For example, if you want to find the square mileage of developed land after 5 years, you can substitute $x = 5$ into the equation:

$$y = 8,500(1.12)^5$$

Simplifying the equation, you get:

$$y = 8,500(1.7628)$$

$$y = 14,943.6$$

Therefore, the square mileage of developed land after 5 years will be approximately 14,943.6 square miles.

A: The growth factor is the rate at which the developed land is increasing. In this case, the growth factor is $1 + 0.12 = 1.12$, which means that the developed land is increasing by $12\%$ per year. The growth factor is an important component of the equation, as it determines the rate at which the developed land is growing.

A: Yes, the equation for exponential growth can be used to model other types of growth, such as population growth, economic growth, and technological growth. The equation is a general model that can be applied to a wide range of situations.

A: The accuracy of the equation depends on the assumptions made about the growth rate and the initial value. In this case, the equation assumes that the growth rate is constant at $12\%$ per year, and that the initial value is 8,500 square miles. If these assumptions are accurate, then the equation should provide a good prediction of the square mileage of developed land. However, if the assumptions are not accurate, then the equation may not provide a good prediction.

A: Yes, the equation can be used to predict the square mileage of developed land in the long term. However, it's worth noting that the equation assumes that the growth rate remains constant over time, which may not be the case in reality. In the long term, the growth rate may change due to various factors such as changes in population growth, economic growth, and technological advancements.

A: Some limitations of the equation include:

• The equation assumes that the growth rate is constant over time, which may not be the case in reality.
• The equation assumes that the initial value is known with certainty, which may not be the case in reality.
• The equation does not take into account other factors that may affect the growth of developed land, such as changes in government policies, technological advancements, and economic conditions.

In conclusion, the equation for exponential growth is a powerful tool for modeling population growth and other types of growth. The equation can be used to predict the square mileage of developed land after a certain number of years, and it can be applied to a wide range of situations. However, it's worth noting that the equation has some limitations, and it should be used with caution.