A radioactive substance decays by 0.2% each year. a) If you started with 90mg of this substance, how much would remain after 100 years? b) What is the half-life of this substance?
A radioactive substance decays by 0.2% each year. a) If you started with 90mg of this substance, how much would remain after 100 years? b) What is the half-life of this substance?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Radioactive Decay Exercise
#### Problem Statement:
A radioactive substance decays by 0.2% each year.
#### Questions:
a) If you started with 90mg of this substance, how much would remain after 100 years?
[Text Box for Answer]
b) What is the half-life of this substance?
[Text Box for Answer]
---
Explanation: To solve part (a), use the exponential decay formula:
\[ A = P(1 - r)^t \]
where:
- \( A \) is the amount that remains after time \( t \).
- \( P \) is the initial amount (90mg in this case).
- \( r \) is the decay rate (0.2% or 0.002 as a decimal).
- \( t \) is the time in years (100 years).
To solve part (b), use the formula for half-life \( T_{1/2} \):
\[ T_{1/2} = \frac{\ln(2)}{\ln(1 + r)} \]
where \( \ln \) is the natural logarithm.
Let's calculate step-by-step in the solution section.
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Transcribed Image Text:---
### Radioactive Decay Exercise
#### Problem Statement:
A radioactive substance decays by 0.2% each year.
#### Questions:
a) If you started with 90mg of this substance, how much would remain after 100 years?
[Text Box for Answer]
b) What is the half-life of this substance?
[Text Box for Answer]
---
Explanation: To solve part (a), use the exponential decay formula:
\[ A = P(1 - r)^t \]
where:
- \( A \) is the amount that remains after time \( t \).
- \( P \) is the initial amount (90mg in this case).
- \( r \) is the decay rate (0.2% or 0.002 as a decimal).
- \( t \) is the time in years (100 years).
To solve part (b), use the formula for half-life \( T_{1/2} \):
\[ T_{1/2} = \frac{\ln(2)}{\ln(1 + r)} \]
where \( \ln \) is the natural logarithm.
Let's calculate step-by-step in the solution section.
---
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