2023 ASRJC P1 Q10

Glucose is a type of sugar in our blood and our body uses it for energy. Having low sugar in the blood for long periods of time can cause health problems if it is not treated.

At 10:00 am, a patient is given glucose via an intravenous drip at a constant rate of \(r\) units of glucose per hour. It is known that the rate at which the glucose in the blood stream is absorbed by the body, is proportional to the amount of glucose present in the blood stream. At the start of the treatment, the patient has \({{x}_{0}}\)units of glucose in his blood stream.

It is given that \(x\) denotes the amount of glucose present in the blood stream of the patient at time \(t\) hours after he starts the treatment.

1. Form a differential equation in \(x\) and show that \(x=\left( {{x}_{0}}-\frac{r}{k} \right){{\mathrm{e}}^{-kt}}+\frac{r}{k}\) where \(k\) is a positive constant.[4]

It is now given that \(\frac{r}{k}=2{{x}_{0}}\).

2. State the theoretical limiting value of \(x\) if the patient is given the treatment over a long period.[1]
3. Sketch the graph of \(x\) for \(t\ge 0\).[2]

The patient has \(10\%\) more glucose in his blood stream after \(\frac{1}{2}\) hour of treatment.

4. At what time (to the nearest minute), would the glucose in his blood stream first exceed \(90\%\) of the value found in part (b)?[5]