2022 ASRJC J2 MYCT P1 Q11

A patient is injected with \(150\) mg of a drug for treatment of a disease. Every \(8\) hours, \(15\%\) of the drug passes out of his bloodstream. To compensate the loss, a further \(x\) mg dose is given every \(8\) hours.

1. Show that the amount of drug (mg) in the patient’s bloodstream immediately after the nth dose is given by \(\left( 150-\frac{20}{3}x \right){{\left( 0.85 \right)}^{n-1}}+\frac{20}{3}x\).[4]

The safety limit of the amount of drug in a person’s bloodstream is \(165\) mg.

2. In the case where \(x=25\), find the maximum number of doses that can be administered to the patient for the amount of drug to not exceed the safety limit.[2]

Another patient is initially injected with \(165\)mg of the drug. The amount of drug present at time \(t\) hours in the bloodstream of the patient is \(V\)mg. The drug is removed naturally from his bloodstream at a rate \(kV\)mg/hour for some positive constant \(k\).

3. Write down and solve a differential equation, giving \(V\) in terms of \(t\) and \(k\).[4]

4. The drug is effective for treatment only if there is at least 85 mg in the body. Given that 𝑉 =100 after 24 hours, find the time taken for the amount of drug to drop from 165 mg to the minimum effective level, giving your answer to the nearest hour.

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