Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem:**
Substance A decomposes at a rate proportional to the amount of A present. It is found that 18 lb of A will reduce to 9 lb in 4.3 hr. After how long will there be only 1 lb left?
---
**Answer Field:**
There will be 1 lb left after [ ] hr.
*(Do not round until the final answer. Then round to the nearest whole number as needed.)*
---
**Explanation:**
This problem involves exponential decay, where the rate of decomposition is proportional to the current amount. This implies the use of the exponential decay formula:
\[ A(t) = A_0 \cdot e^{-kt} \]
where:
- \( A(t) \) is the amount of substance at time \( t \),
- \( A_0 \) is the initial amount of the substance,
- \( k \) is the decay constant,
- \( t \) is the time.
Given:
- \( A_0 = 18 \) lb,
- \( A(t) = 9 \) lb when \( t = 4.3 \) hr.
Using this information, we first determine the decay constant \( k \) and then solve for the time when \( A(t) = 1 \) lb.](https://content.bartleby.com/qna-images/question/514923f2-e704-4cb3-8e16-d4c01e5e97ad/632024db-4a05-4d90-9ada-def5b7fe655a/j8vav1i_processed.png)
Transcribed Image Text:**Problem:**
Substance A decomposes at a rate proportional to the amount of A present. It is found that 18 lb of A will reduce to 9 lb in 4.3 hr. After how long will there be only 1 lb left?
---
**Answer Field:**
There will be 1 lb left after [ ] hr.
*(Do not round until the final answer. Then round to the nearest whole number as needed.)*
---
**Explanation:**
This problem involves exponential decay, where the rate of decomposition is proportional to the current amount. This implies the use of the exponential decay formula:
\[ A(t) = A_0 \cdot e^{-kt} \]
where:
- \( A(t) \) is the amount of substance at time \( t \),
- \( A_0 \) is the initial amount of the substance,
- \( k \) is the decay constant,
- \( t \) is the time.
Given:
- \( A_0 = 18 \) lb,
- \( A(t) = 9 \) lb when \( t = 4.3 \) hr.
Using this information, we first determine the decay constant \( k \) and then solve for the time when \( A(t) = 1 \) lb.
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