2017 AJC P2 Q1

At the intensive care unit of a hospital, patients of a particular condition receive a certain treatment drug through an intravenous drip at a constant rate of \(30\) mg per hour. Due to the limited capacity for absorption by the body, the drug is lost from a patient’s body at a rate proportional to \(x\), where \(x\) is the amount of drug (in mg) present in the body at time \(t\) (in hours). It is assumed that there is no presence of the drug in any patient prior to admission to the hospital.

1. Form a differential equation involving \(x\) and \(t\) and show that \(x=\frac{30}{k}\left( 1-{{e}^{-kt}} \right)\) where \(k\) is a positive constant.[4]
2. If there is more than \(1000\)mg of drug present in a patient’s body, it is considered an overdose. Suppose the drug continues to be administered, determine the range of values of \(k\) such that a patient will have an overdose.[2]

For a particular patient, \(k=\frac{1}{50}\).

3. Find the time required for the amount of the drug present in the patient’s body to be \(200\) mg.

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