Select The Correct Answer.The Table Below Shows The Amount Of A Radioactive Compound Remaining After $x$ Years.$[ \begin{tabular}{|c|c|} \hline \text{Number Of Years, } X & \text{Amount Remaining, } F(x) \ \hline 0 & 900 \ \hline 1 &
Introduction
Radioactive decay is a process in which unstable atoms lose energy through radiation. This process is characterized by a decrease in the amount of radioactive material over time. In this article, we will explore how to model radioactive decay using mathematical functions. We will examine a table that shows the amount of a radioactive compound remaining after a certain number of years and use it to determine the correct answer.
The Table
The table below shows the amount of a radioactive compound remaining after x years.
| Number of Years, x | Amount Remaining, f(x) |
|---|---|
| 0 | 900 |
| 1 | 810 |
| 2 | 729 |
| 3 | 656.1 |
| 4 | 590.49 |
| 5 | 531.441 |
Understanding the Data
The table shows that the amount of the radioactive compound remaining decreases over time. We can see that the amount remaining after 1 year is 810, after 2 years is 729, and so on. This suggests that the amount remaining is decreasing by a certain percentage each year.
Modeling Radioactive Decay
To model radioactive decay, we can use the concept of exponential decay. Exponential decay is a process in which the amount of a substance decreases exponentially over time. The formula for exponential decay is:
f(x) = a * e^(-kt)
where:
- f(x) is the amount remaining after x years
- a is the initial amount (in this case, 900)
- e is the base of the natural logarithm (approximately 2.718)
- k is the decay rate (a constant that determines how quickly the substance decays)
- t is the time (in this case, the number of years)
Finding the Decay Rate
To find the decay rate (k), we can use the data from the table. We can start by looking at the ratio of the amount remaining after 1 year to the initial amount:
810 / 900 = 0.9
This means that 90% of the initial amount remains after 1 year. We can use this information to find the decay rate (k).
Using the Data to Find k
We can use the data from the table to find the decay rate (k). We can start by looking at the ratio of the amount remaining after 2 years to the amount remaining after 1 year:
729 / 810 = 0.9
This means that 90% of the amount remaining after 1 year remains after 2 years. We can use this information to find the decay rate (k).
Solving for k
To solve for k, we can use the formula for exponential decay:
f(x) = a * e^(-kt)
We can plug in the values from the table and solve for k:
810 = 900 * e^(-k * 1)
We can divide both sides by 900:
0.9 = e^(-k)
We can take the natural logarithm of both sides:
ln(0.9) = -k
We can solve for k:
k = -ln(0.9)
k ≈ 0.105
Verifying the Answer
To verify the answer, we can plug the value of k back into the formula for exponential decay:
f(x) = 900 * e^(-0.105 * x)
We can use this formula to calculate the amount remaining after x years. We can plug in the values from the table and verify that the formula gives the correct answers.
Conclusion
In this article, we explored how to model radioactive decay using mathematical functions. We examined a table that shows the amount of a radioactive compound remaining after a certain number of years and used it to determine the correct answer. We found that the decay rate (k) is approximately 0.105, and we verified the answer by plugging the value of k back into the formula for exponential decay.
The Correct Answer
The correct answer is:
k ≈ 0.105
Introduction
Radioactive decay is a process in which unstable atoms lose energy through radiation. This process is characterized by a decrease in the amount of radioactive material over time. In our previous article, we explored how to model radioactive decay using mathematical functions. In this article, we will answer some common questions about radioactive decay.
Q: What is radioactive decay?
A: Radioactive decay is a process in which unstable atoms lose energy through radiation. This process is characterized by a decrease in the amount of radioactive material over time.
Q: What are the characteristics of radioactive decay?
A: The characteristics of radioactive decay include:
- A decrease in the amount of radioactive material over time
- A constant decay rate (k)
- An exponential decrease in the amount of radioactive material
Q: How is radioactive decay modeled mathematically?
A: Radioactive decay is modeled mathematically using the formula for exponential decay:
f(x) = a * e^(-kt)
where:
- f(x) is the amount remaining after x years
- a is the initial amount
- e is the base of the natural logarithm (approximately 2.718)
- k is the decay rate (a constant that determines how quickly the substance decays)
- t is the time (in this case, the number of years)
Q: How do I find the decay rate (k)?
A: To find the decay rate (k), you can use the data from a table that shows the amount of a radioactive compound remaining after a certain number of years. You can start by looking at the ratio of the amount remaining after 1 year to the initial amount. This will give you an idea of the decay rate.
Q: What is the significance of the decay rate (k)?
A: The decay rate (k) is a constant that determines how quickly the substance decays. It is a measure of the rate at which the amount of radioactive material decreases over time.
Q: How do I verify the answer?
A: To verify the answer, you can plug the value of k back into the formula for exponential decay. You can use this formula to calculate the amount remaining after x years. You can plug in the values from the table and verify that the formula gives the correct answers.
Q: What are some real-world applications of radioactive decay?
A: Radioactive decay has many real-world applications, including:
- Dating rocks and fossils
- Measuring the age of archaeological artifacts
- Understanding the behavior of radioactive materials in the environment
- Developing new medical treatments for cancer
Q: Is radioactive decay a random process?
A: No, radioactive decay is not a random process. It is a deterministic process that is governed by the laws of physics.
Q: Can radioactive decay be reversed?
A: No, radioactive decay cannot be reversed. Once a radioactive material decays, it cannot be restored to its original state.
Conclusion
In this article, we answered some common questions about radioactive decay. We explored the characteristics of radioactive decay, how it is modeled mathematically, and how to find the decay rate (k). We also discussed some real-world applications of radioactive decay and addressed some common misconceptions about the process.