The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years? mg Give your answer accurate to at least 2 decimal places.
The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years? mg Give your answer accurate to at least 2 decimal places.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years?
![### Radioactive Decay Calculation Tutorial
**Problem Statement:**
The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years?
[Interactable Text Box]
```plaintext
___ mg
```
*Give your answer accurate to at least 2 decimal places.*
### Explanation and Solution:
The concept of half-life is used to describe the time it takes for a substance to reduce to half its initial amount due to radioactive decay. Here, we use the decay formula:
\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
Where:
- \( N(t) \) = amount remaining after time \( t \)
- \( N_0 \) = initial amount
- \( t \) = elapsed time
- \( T_{1/2} \) = half-life of the substance
Plugging in the values given:
- \( N_0 = 100 \) mg
- \( t = 2000 \) years
- \( T_{1/2} = 1590 \) years
First, compute the exponent:
\[ \frac{t}{T_{1/2}} = \frac{2000}{1590} \approx 1.26 \]
Now apply to the formula:
\[ N(2000) = 100 \left( \frac{1}{2} \right)^{1.26} \approx 100 \times 0.4156 \approx 41.56 \text{ mg} \]
So, approximately 41.56 mg of Radium-226 will remain after 2000 years.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07450324-4898-4505-8a10-20a625d5cb7d%2Fd1aa94bf-ca23-4d9b-a06f-34492db6737f%2Fvl70cha.png&w=3840&q=75)
Transcribed Image Text:### Radioactive Decay Calculation Tutorial
**Problem Statement:**
The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 2000 years?
[Interactable Text Box]
```plaintext
___ mg
```
*Give your answer accurate to at least 2 decimal places.*
### Explanation and Solution:
The concept of half-life is used to describe the time it takes for a substance to reduce to half its initial amount due to radioactive decay. Here, we use the decay formula:
\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
Where:
- \( N(t) \) = amount remaining after time \( t \)
- \( N_0 \) = initial amount
- \( t \) = elapsed time
- \( T_{1/2} \) = half-life of the substance
Plugging in the values given:
- \( N_0 = 100 \) mg
- \( t = 2000 \) years
- \( T_{1/2} = 1590 \) years
First, compute the exponent:
\[ \frac{t}{T_{1/2}} = \frac{2000}{1590} \approx 1.26 \]
Now apply to the formula:
\[ N(2000) = 100 \left( \frac{1}{2} \right)^{1.26} \approx 100 \times 0.4156 \approx 41.56 \text{ mg} \]
So, approximately 41.56 mg of Radium-226 will remain after 2000 years.
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