Question

A worker at a hazardous waste plant was accidentally exposed to toxic chemicals which were absorbed into his bloodstream. Upon feeling ill, the worker went to a hospital and had some blood drawn for testing. The concentration of chemical in the drawn blood was found to be 0.158 mg/ml. Expensive medication was administered to counter the effects of the chemical in the blood, but the doctor on duty knew that the concentration of the chemical in the bloodstream would have to decrease gradually over time according to the Basic Law of Exponential Decay (y = Ce^(-kt)). Medication would have to be administered every hour until the concentration was below 0.050 mg/ml. Two hours later, blood was again drawn, and it was found to contain a chemical concentration of 0.126 mg/ml. The doctor asked a lab technician to do the following: You do the same. Write the particular solution for exponential decay for the chemical in the patient's blood. (Let t = 0 represent the time that blood was first drawn.) Sketch a graph of the function from Part a. Find out how long it will be before the patient can be taken off medication. When the patient has only a negligible amount of chemical in his bloodstream (less than 0.0001 mg/ml), he can be released from the hospital. Find out how long the patient has to be hospitalized (from the time he first came to the hospital and had his blood drawn). Occasionally, patients exposed to this chemical suffer damage to their central nervous systems. A maximum concentration of 0.020 mg/ml requires a follow-up examination. The doctor estimated that the maximum concentration of the chemical in the worker's bloodstream occurred 1 hour after exposure. The patient estimated 4 hours after exposure would have been about 3 hours prior to his blood being drawn for the first time. Should the patient be asked to return for a follow-up exam? Why or why not? Find the half-life for the chemical in the bloodstream for the patient.

Answer 1

We use the exponential decay formula to find the decay constant and the particular solution for a chemical in a patient's bloodstream, determine thresholds for additional medication and discharge, assess the necessity for a follow-up exam and calculate the half-life of the chemical.

Step-by-step explanation:

The situation described involves determining the particular solution for the exponential decay of a chemical in a patient's bloodstream. Since the initial concentration (C) at t = 0 is 0.158 mg/ml and at t = 2 hours, the concentration is 0.126 mg/ml, we can use the exponential decay formula, y = Ce-kt, to find the decay constant (k) and then solve for the particular solution. We can then use the formula to find out how long until the concentration falls below the given thresholds and to determine whether follow-up examination is necessary.

To determine if the patient should be asked to return for a follow-up exam, we need to predict the maximum concentration in the bloodstream. To do this, we assume that the exponential growth happened in the hour prior to the first measurement. Assuming a linear increase in concentration, we must calculate if the concentration at the peak was above 0.020 mg/ml to decide if a follow-up exam is necessary.

Finally, to find the half-life of the chemical in the bloodstream, we need to solve for the time (t) when the initial concentration (C) has been reduced by half using the previously calculated decay constant (k) in the exponential decay formula.