2022 R1 P1 Q10

Glucose in the blood stream is reduced at a rate proportional to the amount of glucose present in the blood stream.

Let \(G\) milligrams (mg) be the amount of glucose in \(1\) decilitre (dL) of blood stream at time \(t\) minutes and let \(\lambda \) denote the positive constant of proportionality.

1. Write down a differential equation relating \(G\), \(\lambda \) and \(t\). Solve this differential equation to find an expression of \(G\) in terms of \(\lambda \) and \(t\).[3]

Through extensive research, doctors recommended that a healthy glucose level in the blood stream of an adult should be between \(70\) mg/dL to \(100\) mg/dL.

An adult patient, Neo, has \(80\) mg/dL of glucose in his blood stream at time \(t=0\) minutes. Due to a medical condition, he is not able to extract glucose from the food he eats. As he is not able to replenish the glucose in his blood stream, he is thus losing his glucose in the blood stream at a rate (in mg/dL per minute) corresponding to \(\lambda =0.005\) per minute.

2. Find the approximate time it would take for Neo’s glucose level in the blood stream to fall below the healthy range if there is no intake of glucose.[2]

In order to maintain Neo’s glucose level in the blood stream in the healthy range, glucose is injected into his blood stream intravenously at a constant rate of \(\mu \) mg/dL per minute.

3. Write down a differential equation relating \(G\), \(\mu \) and \(t\) to model Neo’s situation. Solve this differential equation to find an expression of \(G\) in terms of \(\mu \) and \(t\).[5]
4. Explain whether it is recommended to keep injecting glucose at a rate of \(0.7\) mg/dL per minute into Neo’s blood stream.[2]
5. Find the maximum range of values of \(\mu \) so that Neo’s glucose level in the blood stream will always be in the healthy range.[1]