a. Find the fraction of lead in the system that will be in the blood and in the tissues in the long run . b. Find a formula that gives the amount Xn and Yn for any month . Since you do not have an initial condition, you will find equations that measure the amounts for any starting value.
a. Find the fraction of lead in the system that will be in the blood and in the tissues in the long run . b. Find a formula that gives the amount Xn and Yn for any month . Since you do not have an initial condition, you will find equations that measure the amounts for any starting value.
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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![### Understanding Matrix Models in Describing Toxin Flow
Matrix models can be used to describe the flow of certain toxins, such as lead, between compartments of a biological system. For example, lead enters the human body by inhaling or ingesting lead particles. These particles are diffused into the tissue such as the kidney and liver and into the bloodstream.
Let \( X_n \) represent the amount of lead in the blood (in micrograms) and let \( Y_n \) represent the amount in the tissue at the start of a certain month.
A model for calculating the levels a month later is given by:
\[ \begin{pmatrix} X_{n+1} \\ Y_{n+1} \end{pmatrix} = \begin{pmatrix} 0.39 & 0.04 \\ 0.04 & 0.02 \end{pmatrix} \begin{pmatrix} X_n \\ Y_n \end{pmatrix} \]
#### Questions to Answer:
- Use the next blank page for your work. Only write the answers here.
a. Find the fraction of lead in the system that will be in the blood and in the tissues in the long run.
b. Find a formula that gives the amount \( X_n \) and \( Y_n \) for any month. Since you do not have an initial condition, you will find equations that measure the amounts for any starting value.
\[ X_n = \underline{\hspace{15cm}} \]
\[ Y_n = \underline{\hspace{15cm}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97ffc8ee-2daf-45a3-bf2a-1bfa124f857e%2F7383468f-23de-4c46-a14a-9bf9e979e3d2%2Fwg4sdg_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Matrix Models in Describing Toxin Flow
Matrix models can be used to describe the flow of certain toxins, such as lead, between compartments of a biological system. For example, lead enters the human body by inhaling or ingesting lead particles. These particles are diffused into the tissue such as the kidney and liver and into the bloodstream.
Let \( X_n \) represent the amount of lead in the blood (in micrograms) and let \( Y_n \) represent the amount in the tissue at the start of a certain month.
A model for calculating the levels a month later is given by:
\[ \begin{pmatrix} X_{n+1} \\ Y_{n+1} \end{pmatrix} = \begin{pmatrix} 0.39 & 0.04 \\ 0.04 & 0.02 \end{pmatrix} \begin{pmatrix} X_n \\ Y_n \end{pmatrix} \]
#### Questions to Answer:
- Use the next blank page for your work. Only write the answers here.
a. Find the fraction of lead in the system that will be in the blood and in the tissues in the long run.
b. Find a formula that gives the amount \( X_n \) and \( Y_n \) for any month. Since you do not have an initial condition, you will find equations that measure the amounts for any starting value.
\[ X_n = \underline{\hspace{15cm}} \]
\[ Y_n = \underline{\hspace{15cm}} \]
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