### Compartment Analysis: Drug Concentration in Blood and Cerebrospinal Fluid#### Problem Statement:A patient initially has a concentration of 50 mg/L of morphine in her blood and 70 mg/L in her cerebrospinal fluid. Suppose that the drug moves into the cerebrospinal fluid at a per-capita rate of 0.4 and that the amount that is moved back from the cerebrospinal fluid to the bloodstream is 0.2. #### Metabolism Assumptions:Further suppose that the metabolism of medicine in the bloodstream is 0.1 and the metabolism of the drug in the cerebrospinal fluid is 0.2.#### Compartment Diagram:The diagram is divided into two compartments:1. **Blood**2. **Cerebrospinal Fluid (Brain)**(Note: The provided image contains two empty boxes here that students need to fill as part of the exercise.)#### Variables:Let \( x(t) \) represent the concentration of the drug in the blood.Let \( y(t) \) represent the concentration of the drug in the cerebrospinal fluid (brain area).Complete this compartment diagram.#### Explanation of the Mathematical Model:The given differential equations represent the rates of change in drug concentration in the blood and cerebrospinal fluid based on the given metabolism and transfer rates.##### Differential Equations:\[\frac{dx}{dt} = -0.5x + 0.2y\]\[\frac{dy}{dt} = 0.4x - 0.4y\]#### Task:Solve for \( x(t) \) and \( y(t) \) given the initial conditions stated above. Simplify each expression as much as possible and write them in the blanks below.- **Initial Concentrations:** - Blood: \( x(0) = 50 \) mg/L - Cerebrospinal Fluid: \( y(0) = 70 \) mg/L##### Solutions:X(t) = __________________________________________________________Y(t) = __________________________________________________________**Note:** On the next page, please show all your calculations.---By analyzing and solving these differential equations, we can understand how the morphine concentration changes over time in the patient's bloodstream and cerebrospinal fluid, taking into account both metabolism and inter-compartment transfer rates.

5. A patient initially has a concentration of 50mg/L of morphine in her blood and 70mg/L in her cerebrospinal fluid. Suppose that the drug moves into the cerebrospinal fluid at a per-capita rate of .4 and that the amount that is moved back from the cerebrospinal fluid to the bloodstream is .2 . Suppose further that the metabolism of medicine in the bloodstream is .1 and the metabolism of the drug in the cerebrospinal fluid is .2 Blood cerebrospinal fluid ( brain ) Let x(t) represent the concentration in the blood and y(t) represent the concentration in the

5. A patient initially has a concentration of 50mg/L of morphine in her blood and 70mg/L in her cerebrospinal fluid. Suppose that the drug moves into the cerebrospinal fluid at a per-capita rate of .4 and that the amount that is moved back from the cerebrospinal fluid to the bloodstream is .2 . Suppose further that the metabolism of medicine in the bloodstream is .1 and the metabolism of the drug in the cerebrospinal fluid is .2 Blood cerebrospinal fluid ( brain ) Let x(t) represent the concentration in the blood and y(t) represent the concentration in the

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter5: A Survey Of Other Common Functions

Section5.4: Combining And Decomposing Functions

Problem 18E

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Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if her glucose level is above 155 milligrams per deciliter (mg/dl) one hour after a sugary drink. Sheila's measured glucose level one hour after the sugary drink varies according to the Normal distribution with μ = 140 mg/dl and σ = 11 mg/dl.Let X = Sheila's measured glucose level one hour after a sugary drink(a) P(X > 155) = (Use 3 decimal places)Suppose measurements are made on 3 separate days and the mean result is compared with the criterion 155 mg/dl.(b) P(X > 155) = (Use 3 decimal places)(c) What sample mean blood glucose level is higher than 95% of all other sample mean blood glucose levels? Hint: this requires a backward Normal calculation. (Use 2 decimal places)
Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 127 mg/dl and standard deviation 10 mg/dl. Let L denote a patient's glucose level.
Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 127 mg/dl and standard deviation 10 mg/dl. Let L denote a patient's glucose level.
Shelia's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter one hour after a sugary drink is ingested. Shelia's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with mean 128 mg/dl and standard deviation 9 mg/dl. Let ? denote a patient's glucose level. (a) If measurements are made on four different days, find the level ? such that there is probability only 0.01 that the mean glucose level of four test results falls above ? for Shelia's glucose level distribution. What is the value of ??ANSWER: (b) If the mean result from the four tests is compared to the criterion 140 mg/dl, what is the probability that Shelia is diagnosed as having gestational…
A patient is classified as having gestational diabetes if their average glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Rebecca's doctor is concerned that she may suffer from gestational diabetes. There is variation both in the actual glucose level and in the blood test that measures the level. Rebecca's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ=146 mg/dl and σ=7 mg/dl. Using the Central Limit Theorem, determine the probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured: n=1 n=8 n=10 Comment on the relationship between the probabilities observed in (a), (b), and (c). Explain, using concepts from lecture why this occurs and what it means in context.
A patient is classified as having gestational diabetes if their average glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Rebecca's doctor is concerned that she may suffer from gestational diabetes. There is variation both in the actual glucose level and in the blood test that measures the level. Rebecca's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ=147 mg/dl and σ=8 mg/dl, Using the Central Limit Theorem, determine the probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured: Once? n=#+2 times, where # is the last digit of your student ID? n=#+4 times, where # is the last digit of your student ID? Comment on the relationship between the probabilities observed in (a), (b), and (c). Explain, using concepts from lecture why this occurs and what it means in context.
How to formulate the model of blood loss?