Explore aspects of what happens to the concentration of Voltar in a patient’s system using a three-dose model.

Scenario:
Voltar is a strong pain relieving medication. The instructions on the packet state that it should be taken at most once every six hours.
The amount of drug in the bloodstream is measured in units called parts per million or ppm.
If the concentration of Voltar in the bloodstream rises above 5 ppm it becomes dangerous. Voltar is only effective at relieving pain if the concentration in the bloodstream is above 1.1 ppm.

Part A – The surge function.

A surge function is in the form where A and b are positive constants.
On the same axes, graph and for the case where and . ie
Determine the coordinates of the stationary point and point of inflection and label these on the graph.
Repeat the investigation for three different values of while maintaining .
Include your graphs in the report and summarise the findings in a suitable table.
State the effect of changing the value of on the graph of .
Using a similar process investigate the effect of changing the value of on the graph of
.
Make a conjecture on how the value of b effects the x-coordinates of the stationary point and the point of inflection of the graph of .
Prove your conjecture.
Comment on the suitability of the surge function in modelling medicinal doses by relating the features of the graph to the effect that a medicinal dose has on the body.
Discuss any limitations of the model.
At least four key points should be made.

Part B – The basic surge model.

The concentration of Voltar in a patient’s blood stream t hours after a single dose has been taken, in parts per million (ppm), is given by the function

Draw the graph of for the first 6 hours after a single dose has been taken. On your graph label the dangerous level of 5 ppm and the effective level of 1.1 ppm.
Between what times after the dose is taken is the medicine Voltar effective? What percentage of the first six hours is it effective?
Find the exact value of the maximum concentration over the six hours using calculus. Does the drug Voltar ever reach dangerous levels?
Use calculus to find the point at which the rate the drug is leaving the person’s system is at a maximum. At what rate is it leaving the system at this time?
Explain why it should be safe to take a second dose of Voltar after 6 hours have passed.

Part C – The two-dose surge model.
After four hours, a patient takes a second dose of Voltar.
Explain why the concentration of the medication Voltar in the patient’s system can be given by:

Draw a graph of the concentration of the medication in the patient’s bloodstream using the functions directly above for the first 12 hours after the first dose is taken. Label the maximum points on the graph.
Find using calculus, if the concentration of Voltar in the bloodstream ever reaches dangerous levels.
A patient takes one dose and then a second dose an hour later. Examine using algebraic and/or graphical methods if this will be dangerous.
Use algebraic and/or graphical methods to investigate the earliest time (accurate to two decimal places) that a patient can safely take a second dose after the first.

Part D – Multiple-dose surge models
A patient suffers a sporting injury and needs to take multiple doses of Voltar to manage the pain over a longer period of time.
Explore aspects of what happens to the concentration of Voltar in a patient’s system using a three-dose model.
Discuss any assumption and limitations used in this model.

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