2. The half-life of a radioactive substance is 4000 years. How long will it take for the substance to reduce its size to 1/3 of the present amount?
2. The half-life of a radioactive substance is 4000 years. How long will it take for the substance to reduce its size to 1/3 of the present amount?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem: Radioactive Decay**
The half-life of a radioactive substance is 4000 years. How long will it take for the substance to reduce its size to 1/3 of the present amount?
**Explanation:**
In this problem, we are dealing with the concept of half-life, which is the time it takes for a quantity to reduce to half its initial amount. The problem requires calculating how long it will take for the substance's quantity to reduce to one-third of its current amount. To solve this, you can use the formula for exponential decay related to half-life:
\[ N(t) = N_0 \left(\frac{1}{2}\right)^{t/T} \]
where:
- \( N(t) \) is the remaining quantity of the substance after time \( t \),
- \( N_0 \) is the initial quantity,
- \( T \) is the half-life of the substance.
You need to find \( t \) when \( N(t) = \frac{1}{3}N_0 \). Solving this equation will give you the time required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d410cd6-8fc6-4d13-8ff0-7a43ba3056bc%2F0c582266-3e83-45c6-8d23-1848d5e9378e%2Ffgo2vgd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Radioactive Decay**
The half-life of a radioactive substance is 4000 years. How long will it take for the substance to reduce its size to 1/3 of the present amount?
**Explanation:**
In this problem, we are dealing with the concept of half-life, which is the time it takes for a quantity to reduce to half its initial amount. The problem requires calculating how long it will take for the substance's quantity to reduce to one-third of its current amount. To solve this, you can use the formula for exponential decay related to half-life:
\[ N(t) = N_0 \left(\frac{1}{2}\right)^{t/T} \]
where:
- \( N(t) \) is the remaining quantity of the substance after time \( t \),
- \( N_0 \) is the initial quantity,
- \( T \) is the half-life of the substance.
You need to find \( t \) when \( N(t) = \frac{1}{3}N_0 \). Solving this equation will give you the time required.
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The half life of the radioactive decay is 
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