An Element With A Mass Of 670 Grams Decays By $27.3 \%$ Per Minute. How Much Of The Element Remains After 9 Minutes, To The Nearest Tenth Of A Gram?
Introduction
Radioactive decay is a process in which unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. This process is characterized by a decrease in the number of radioactive atoms over time, resulting in a decrease in the mass of the substance. In this article, we will explore how to calculate the remaining mass of an element after a certain period of time, given its initial mass and the rate of decay.
Calculating the Remaining Mass
To calculate the remaining mass of the element after a certain period of time, we can use the formula for exponential decay:
A(t) = A0 * (1 - r)^t
where:
- A(t) is the remaining mass at time t
- A0 is the initial mass
- r is the rate of decay (as a decimal)
- t is the time elapsed
In this case, the initial mass (A0) is 670 grams, the rate of decay (r) is 27.3% or 0.273 (as a decimal), and the time elapsed (t) is 9 minutes.
Applying the Formula
We can now plug in the values into the formula:
A(9) = 670 * (1 - 0.273)^9
Calculating the Remaining Mass
To calculate the remaining mass, we need to evaluate the expression (1 - 0.273)^9. This can be done using a calculator or a computer program.
(1 - 0.273)^9 ≈ 0.142
Now, we can multiply this value by the initial mass (670 grams) to get the remaining mass:
A(9) ≈ 670 * 0.142 ≈ 95.06 grams
Rounding to the Nearest Tenth
The problem asks us to round the remaining mass to the nearest tenth of a gram. Therefore, we round 95.06 grams to 95.1 grams.
Conclusion
In this article, we calculated the remaining mass of an element after 9 minutes, given its initial mass and the rate of decay. We used the formula for exponential decay and applied it to the given values. The remaining mass was calculated to be approximately 95.1 grams, rounded to the nearest tenth of a gram.
Example Use Case
This problem can be applied to various real-world scenarios, such as:
- Calculating the remaining amount of a radioactive substance in a nuclear reactor
- Determining the shelf life of a radioactive material
- Estimating the amount of radiation emitted by a radioactive source
Tips and Variations
- To calculate the remaining mass after a different time period, simply plug in the new value of t into the formula.
- To calculate the rate of decay (r) from the remaining mass and time elapsed, rearrange the formula to solve for r.
- To calculate the initial mass (A0) from the remaining mass and time elapsed, rearrange the formula to solve for A0.
References
- [1] Radioactive Decay Formula. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Radioactive_decay#Formula
- [2] Exponential Decay. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/exponential-decay.html
Further Reading
- Radioactive Decay: A Comprehensive Guide
- Exponential Decay: Applications and Examples
- Calculating Radioactive Decay: A Step-by-Step Guide
Introduction
Radioactive decay is a process in which unstable atomic nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. This process is characterized by a decrease in the number of radioactive atoms over time, resulting in a decrease in the mass of the substance. In this article, we will explore how to calculate the remaining mass of an element after a certain period of time, given its initial mass and the rate of decay.
Calculating the Remaining Mass
To calculate the remaining mass of the element after a certain period of time, we can use the formula for exponential decay:
A(t) = A0 * (1 - r)^t
where:
- A(t) is the remaining mass at time t
- A0 is the initial mass
- r is the rate of decay (as a decimal)
- t is the time elapsed
In this case, the initial mass (A0) is 670 grams, the rate of decay (r) is 27.3% or 0.273 (as a decimal), and the time elapsed (t) is 9 minutes.
Applying the Formula
We can now plug in the values into the formula:
A(9) = 670 * (1 - 0.273)^9
Calculating the Remaining Mass
To calculate the remaining mass, we need to evaluate the expression (1 - 0.273)^9. This can be done using a calculator or a computer program.
(1 - 0.273)^9 ≈ 0.142
Now, we can multiply this value by the initial mass (670 grams) to get the remaining mass:
A(9) ≈ 670 * 0.142 ≈ 95.06 grams
Rounding to the Nearest Tenth
The problem asks us to round the remaining mass to the nearest tenth of a gram. Therefore, we round 95.06 grams to 95.1 grams.
Conclusion
In this article, we calculated the remaining mass of an element after 9 minutes, given its initial mass and the rate of decay. We used the formula for exponential decay and applied it to the given values. The remaining mass was calculated to be approximately 95.1 grams, rounded to the nearest tenth of a gram.
Q&A
Q: What is the formula for exponential decay?
A: The formula for exponential decay is A(t) = A0 * (1 - r)^t, where A(t) is the remaining mass at time t, A0 is the initial mass, r is the rate of decay (as a decimal), and t is the time elapsed.
Q: What is the rate of decay (r) in this problem?
A: The rate of decay (r) is 27.3% or 0.273 (as a decimal).
Q: What is the initial mass (A0) in this problem?
A: The initial mass (A0) is 670 grams.
Q: What is the time elapsed (t) in this problem?
A: The time elapsed (t) is 9 minutes.
Q: How do I calculate the remaining mass after a different time period?
A: To calculate the remaining mass after a different time period, simply plug in the new value of t into the formula A(t) = A0 * (1 - r)^t.
Q: How do I calculate the rate of decay (r) from the remaining mass and time elapsed?
A: To calculate the rate of decay (r) from the remaining mass and time elapsed, rearrange the formula to solve for r: r = 1 - (A(t)/A0)^(1/t).
Q: How do I calculate the initial mass (A0) from the remaining mass and time elapsed?
A: To calculate the initial mass (A0) from the remaining mass and time elapsed, rearrange the formula to solve for A0: A0 = A(t) / (1 - r)^t.
Q: What is the significance of radioactive decay in real-world applications?
A: Radioactive decay has significant implications in various fields, including nuclear energy, medicine, and environmental science. It is used to calculate the remaining amount of a radioactive substance in a nuclear reactor, determine the shelf life of a radioactive material, and estimate the amount of radiation emitted by a radioactive source.
Further Reading
- Radioactive Decay: A Comprehensive Guide
- Exponential Decay: Applications and Examples
- Calculating Radioactive Decay: A Step-by-Step Guide