January 10, 2008Parkview CampusKalamazoo MI 49008Dear Students of Math 374,Here at Brongess Hospital, we are experts at hip replacement surgery. We have not lost a patient in 10 years, and we want to maintain our excellent record.During surgery, patients lose blood. We have two primary concerns. First, the heart can be harmed by a change in fluid volume in the circulatory system. We are careful to estimate the fluid loss, and we inject a fluid into the bloodstream to replace it. Second, the level of red blood cells must also be maintained. We measure hemocrit, or the percentage of blood volume taken up by red blood cells. A hemocrit level of less than 25% is likely fatal. Before surgery, the average patient has about 5 L of blood which is about 46% hemocrit. A standard hip replacement surgery takes 5 hours, and the average patient loses about 2.5 L of fluid at a constant rate during the surgery. We have been replacing this fluid with a saline solution to maintain blood volume.As I mentioned in my opening paragraph, we have an excellent record, and we would like to maintain it. We now have cause for concern. One of my colleagues has pointed out that if our patients lose 2.5 L of blood at a constant rate for five hours, then they lose LUkiKkclKnByb3RlY3RlZEc2JCQiI0QhIiIjIiIiIiImprint(); # input placeholderL per hour. That is a large fraction of the 5L of blood in the bloodstream: LUkiKkclKnByb3RlY3RlZEc2JCQiIiYhIiIjIiIiRic=print(); # input placeholderThen he argues, our patients should lose that same fraction of hemocrit. So every hour, they should lose LUkiKkclKnByb3RlY3RlZEc2JCQiIiIhIiIiI1k=print(); # input placeholderpercent of the hemocrit in their body. That is, LyomSSNkUEc2IiIiIkkjZHRHRiUhIiIkISNZRig= where P is the percent hemocrit in the blood, and t is time measured in hours. Then he solves this differential equation as follows. He defines the differential equation Pkkkb2RlRzYiLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJQR0YkNiNJInRHRiRGLSQhI1khIiI=print(); # input placeholderThen he finds a general solution to the differential equation LUknZHNvbHZlRzYiNiNJJG9kZUdGJA==print(); # input placeholderThen he defines the initial conditions PkkkaWNzRzYiLy1JIlBHRiQ2IyIiISIjWQ==print(); # input placeholderSo the particular solution is given by LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiM8JEkkb2RlR0YnSSRpY3NHRic=print(); # input placeholderNow here's the disturbing part. When he plots the percentage hemocrit and the critical level of hemocrit on the same axes using LUklcGxvdEc2IjYkPCQiI0QsJkkidEdGJCMhI0IiIiYiI1kiIiIvRik7IiIhRiw=%;the hemocrit level drops below the critical level before the end of the operation! We are hoping you can help us understand our situation. What is wrong with his argument? Can you use differential equations to help us understand why we have been successful with our operations? Using your analysis, can you tell us how long we can allow an operation to run before the hemocrit reaches the critical level?Since our colleague introduced this disturbing analysis, we have started encouraging our patients to donate their own blood. They donate 0.5 L blood a few weeks before the operation. By the operation day, their bodies have rebuilt their blood supply to predonation levels. We would like to be more cautious and keep our patients hemocrit levels higher. In some operations, we have used the blood to replace the fluid lost at the beginning of the operation by blood, and in others we have used it to replace fluid lost at the end of the operation by blood. How would you recommend we use the donated blood to keep our patient's hemocrit levels highest? We look forward to hearing your response by January 24 at 5pm. We would like to see your response in the form of a Maple document. Please send your response to your instructor, and she will forward them to us.Sincerely,Doolb Esaesid