Write An Exponential Growth Model For The Situation:- Initial Value: 2,000- Growth Rate: 6%Choose The Correct Model:A. F ( X ) = 2 , 000 ⋅ 0.06 X F(x) = 2,000 \cdot 0.06^x F ( X ) = 2 , 000 ⋅ 0.0 6 X B. F ( X ) = 2 , 000 ⋅ 106 X F(x) = 2,000 \cdot 106^x F ( X ) = 2 , 000 ⋅ 10 6 X C. F ( X ) = 1.06 ⋅ 2 , 000 X F(x) = 1.06 \cdot 2,000^x F ( X ) = 1.06 ⋅ 2 , 00 0 X D. $f(x) =

Introduction

Exponential growth is a fundamental concept in mathematics, economics, and various other fields. It describes a situation where a quantity increases at a rate proportional to its current value. In this article, we will explore the concept of exponential growth and develop a model to describe a given situation.

Understanding Exponential Growth

Exponential growth can be represented by the equation:

$$y = ab^x$$

where:

• $y$ is the final value
• $a$ is the initial value
• $b$ is the growth rate
• $x$ is the time period

Choosing the Correct Model

We are given the following situation:

• Initial value: 2,000
• Growth rate: 6%

We need to choose the correct model from the given options:

A. $f(x) = 2,000 \cdot 0.06^x$ B. $f(x) = 2,000 \cdot 106^x$ C. $f(x) = 1.06 \cdot 2,000^x$ D. $f(x) = 2,000 \cdot (1.06)^x$

Analyzing the Options

Let's analyze each option to determine the correct model.

Option A: $f(x) = 2,000 \cdot 0.06^x$

This option represents a situation where the growth rate is 0.06, which is 6% of the initial value. However, this option is incorrect because the growth rate should be expressed as a power of 10, not a decimal.

Option B: $f(x) = 2,000 \cdot 106^x$

This option represents a situation where the growth rate is 106, which is not a valid growth rate. The growth rate should be expressed as a power of 10, not a number greater than 10.

Option C: $f(x) = 1.06 \cdot 2,000^x$

This option represents a situation where the growth rate is 1.06, which is 6% more than the initial value. However, this option is incorrect because the growth rate should be expressed as a power of 10, not a decimal.

Option D: $f(x) = 2,000 \cdot (1.06)^x$

This option represents a situation where the growth rate is 1.06, which is 6% more than the initial value. This option is correct because the growth rate is expressed as a power of 10, and the initial value is multiplied by the growth rate raised to the power of x.

Conclusion

In conclusion, the correct model for the given situation is:

$$f(x) = 2,000 \cdot (1.06)^x$$

This model represents a situation where the initial value is 2,000, and the growth rate is 6% per time period.

Example Use Case

Suppose we want to calculate the value of the model after 5 time periods. We can plug in x = 5 into the model:

$$f(5) = 2,000 \cdot (1.06)^5$$

Using a calculator, we get:

$$f(5) = 2,000 \cdot 1.338225 = 2,676.45$$

Therefore, the value of the model after 5 time periods is approximately 2,676.45.

Conclusion

In this article, we developed an exponential growth model to describe a given situation. We analyzed the options and determined that the correct model is:

$$f(x) = 2,000 \cdot (1.06)^x$$

Q&A: Exponential Growth Model

Q: What is exponential growth?

A: Exponential growth is a situation where a quantity increases at a rate proportional to its current value. It can be represented by the equation:

$$y = ab^x$$

where:

• $y$ is the final value
• $a$ is the initial value
• $b$ is the growth rate
• $x$ is the time period

Q: What is the difference between exponential growth and linear growth?

A: Exponential growth and linear growth are two different types of growth patterns. Linear growth is a situation where a quantity increases at a constant rate, whereas exponential growth is a situation where a quantity increases at a rate proportional to its current value.

Q: How do I choose the correct model for exponential growth?

A: To choose the correct model for exponential growth, you need to identify the initial value, growth rate, and time period. The correct model is represented by the equation:

$$f(x) = a \cdot b^x$$

where:

• $a$ is the initial value
• $b$ is the growth rate
• $x$ is the time period

Q: What is the significance of the growth rate in exponential growth?

A: The growth rate is a critical component of exponential growth. It determines the rate at which the quantity increases. A higher growth rate results in a faster increase in the quantity, while a lower growth rate results in a slower increase.

Q: How do I calculate the value of the model after a given time period?

A: To calculate the value of the model after a given time period, you need to plug in the time period into the model. For example, if the model is:

$$f(x) = 2,000 \cdot (1.06)^x$$

and you want to calculate the value after 5 time periods, you can plug in x = 5:

$$f(5) = 2,000 \cdot (1.06)^5$$

Using a calculator, you get:

$$f(5) = 2,000 \cdot 1.338225 = 2,676.45$$

Therefore, the value of the model after 5 time periods is approximately 2,676.45.

Q: What are some real-world applications of exponential growth?

A: Exponential growth has numerous real-world applications, including:

• Population growth: Exponential growth can be used to model population growth, where the population increases at a rate proportional to its current size.
• Financial growth: Exponential growth can be used to model financial growth, where investments or savings increase at a rate proportional to their current value.
• Chemical reactions: Exponential growth can be used to model chemical reactions, where the concentration of a substance increases at a rate proportional to its current concentration.

Q: What are some common mistakes to avoid when working with exponential growth?

A: Some common mistakes to avoid when working with exponential growth include:

• Confusing exponential growth with linear growth
• Using the wrong growth rate or initial value
• Failing to account for time periods or other variables
• Not using a calculator or software to perform calculations

Conclusion

In this article, we provided a comprehensive guide to exponential growth, including a Q&A section. We discussed the concept of exponential growth, how to choose the correct model, and how to calculate the value of the model after a given time period. We also highlighted some real-world applications of exponential growth and common mistakes to avoid.