A Coin Sold For $\$ 214$ In 1976 And Was Sold Again In 1985 For $\$ 460$[/tex\]. Assume That The Growth In The Value \[$ V \$\] Of The Collector's Item Was Exponential.a) Find The Value \[$ K \$\] Of The

Introduction

In the world of collectibles, the value of rare items can fluctuate significantly over time. A coin sold for $214 in 1976 and was sold again in 1985 for $460. This raises an interesting question: what was the rate of growth in the value of this collector's item? In this article, we will explore the concept of exponential growth and use it to determine the value of the coin in 1985.

Exponential Growth

Exponential growth is a type of growth where the rate of change is proportional to the current value. Mathematically, this can be represented as:

$$V(t) = V_0e^{kt}$$

where:

• $V(t)$ is the value at time $t$
• $V_0$ is the initial value
• $k$ is the growth rate
• $t$ is time

The Problem

We are given that the coin was sold for $214 in 1976 and was sold again in 1985 for $460. We want to find the growth rate $k$.

Step 1: Define the Variables

Let's define the variables:

• $V_0 = 214$ (initial value in 1976)
• $V(1985) = 460$ (value in 1985)
• $t = 9$ (time in years, from 1976 to 1985)

Step 2: Plug in the Values

Now, let's plug in the values into the exponential growth equation:

$$460 = 214e^{k(9)}$$

Step 3: Solve for k

To solve for $k$, we can use the following steps:

1. Divide both sides by 214:

$$\frac{460}{214} = e^{9k}$$

2. Take the natural logarithm of both sides:

$$\ln\left(\frac{460}{214}\right) = 9k$$

3. Divide both sides by 9:

$$k = \frac{1}{9}\ln\left(\frac{460}{214}\right)$$

Step 4: Calculate k

Now, let's calculate the value of $k$:

$$k = \frac{1}{9}\ln\left(\frac{460}{214}\right)$$

$$k = \frac{1}{9}\ln(2.15)$$

$$k = \frac{1}{9}(0.763)$$

$$k = 0.085$$

Conclusion

In this article, we used the concept of exponential growth to determine the growth rate of a collector's item. We found that the growth rate $k$ is approximately 0.085. This means that the value of the coin increased by 8.5% per year.

The Significance of Exponential Growth

Exponential growth is a powerful concept that can be used to model a wide range of phenomena, from population growth to financial markets. In the context of collectibles, exponential growth can help us understand the value of rare items over time.

Real-World Applications

Exponential growth has many real-world applications, including:

• Finance: Exponential growth can be used to model the growth of investments, such as stocks and bonds.
• Biology: Exponential growth can be used to model the growth of populations, such as bacteria and viruses.
• Economics: Exponential growth can be used to model the growth of economies, such as GDP and inflation.

Limitations of Exponential Growth

While exponential growth is a powerful concept, it has some limitations. For example:

• Assumes constant growth rate: Exponential growth assumes that the growth rate remains constant over time.
• Does not account for external factors: Exponential growth does not account for external factors, such as changes in market conditions or government policies.

Future Research Directions

Future research directions in exponential growth include:

• Developing more accurate models: Developing more accurate models of exponential growth that account for external factors.
• Applying exponential growth to new fields: Applying exponential growth to new fields, such as medicine and social sciences.

Conclusion

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Q&A: Exponential Growth and Collector's Items

In our previous article, we explored the concept of exponential growth and used it to determine the growth rate of a collector's item. In this article, we will answer some frequently asked questions about exponential growth and collector's items.

Q: What is exponential growth?

A: Exponential growth is a type of growth where the rate of change is proportional to the current value. Mathematically, this can be represented as:

$$V(t) = V_0e^{kt}$$

where:

• $V(t)$ is the value at time $t$
• $V_0$ is the initial value
• $k$ is the growth rate
• $t$ is time

Q: How does exponential growth apply to collector's items?

A: Exponential growth can be used to model the growth of collector's items over time. For example, if a coin is sold for $214 in 1976 and is sold again in 1985 for $460, we can use exponential growth to determine the growth rate of the coin.

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

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

• Finance: Exponential growth can be used to model the growth of investments, such as stocks and bonds.
• Biology: Exponential growth can be used to model the growth of populations, such as bacteria and viruses.
• Economics: Exponential growth can be used to model the growth of economies, such as GDP and inflation.

Q: What are some limitations of exponential growth?

A: While exponential growth is a powerful concept, it has some limitations. For example:

• Assumes constant growth rate: Exponential growth assumes that the growth rate remains constant over time.
• Does not account for external factors: Exponential growth does not account for external factors, such as changes in market conditions or government policies.

Q: How can I apply exponential growth to my own collector's items?

A: To apply exponential growth to your own collector's items, you can use the following steps:

1. Determine the initial value: Determine the initial value of your collector's item.
2. Determine the growth rate: Determine the growth rate of your collector's item using the exponential growth equation.
3. Calculate the future value: Calculate the future value of your collector's item using the exponential growth equation.

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

A: Some common mistakes to avoid when using exponential growth include:

• Assuming a constant growth rate: Exponential growth assumes that the growth rate remains constant over time. However, in reality, growth rates can change over time.
• Not accounting for external factors: Exponential growth does not account for external factors, such as changes in market conditions or government policies.

Q: How can I stay up-to-date with the latest developments in exponential growth?

A: To stay up-to-date with the latest developments in exponential growth, you can:

• Follow reputable sources: Follow reputable sources, such as academic journals and industry publications.
• Attend conferences and workshops: Attend conferences and workshops to learn from experts in the field.
• Join online communities: Join online communities, such as forums and social media groups, to connect with others who are interested in exponential growth.

Conclusion

In conclusion, exponential growth is a powerful concept that can be used to model a wide range of phenomena, including collector's items. By understanding the concept of exponential growth and its limitations, you can make informed decisions about your own collector's items and stay up-to-date with the latest developments in the field.