For a decaying population (K > 0): dN -KN, dt Find the time 7 at which only half of the original population remains.
For a decaying population (K > 0): dN -KN, dt Find the time 7 at which only half of the original population remains.
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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![### Mathematical Modeling of Population Decay
#### Problem Statement:
1. For a decaying population (\(K > 0\)):
\[
\frac{dN}{dt} = -KN,
\]
Find the time \(\tau_{\frac{1}{2}}\) at which only half of the original population remains.
---
This problem involves understanding how a population decreases over time due to decay. The differential equation \(\frac{dN}{dt} = -KN\) models this decay, where \(N\) is the population size, \(t\) is time, and \(K\) is a positive constant representing the decay rate. The task is to determine the half-life (\(\tau_{\frac{1}{2}}\)), which is the time required for the population to reduce to half its initial size.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11d9adce-31d3-450e-a738-16d52e01b365%2Fcba8666c-cf43-4b38-aa42-9d0494261890%2F0y8k9au_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Modeling of Population Decay
#### Problem Statement:
1. For a decaying population (\(K > 0\)):
\[
\frac{dN}{dt} = -KN,
\]
Find the time \(\tau_{\frac{1}{2}}\) at which only half of the original population remains.
---
This problem involves understanding how a population decreases over time due to decay. The differential equation \(\frac{dN}{dt} = -KN\) models this decay, where \(N\) is the population size, \(t\) is time, and \(K\) is a positive constant representing the decay rate. The task is to determine the half-life (\(\tau_{\frac{1}{2}}\)), which is the time required for the population to reduce to half its initial size.
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