When doctors prescribe medicine, they must consider how much the drug’s effectiveness will decrease as time passes. If each hour a drug is 20% less effective as the previous hour, at some point the patient will not be receiving enough medicine and must be given another dose.
A patient was given an initial dose of 200 mg of medication. Write an equation that shows the relationship between the amount of medication remaining in the patient’s bloodstream, and the time in hours, , since the medication was administered.
To write an equation that shows the relationship between the amount of medication remaining in the patient's bloodstream and the time in hours, we need to consider the decrease in effectiveness of the drug over time.
Let's start by defining some variables:
- Let R(t) be the amount of medication remaining in the patient's bloodstream after t hours.
- Let D(t) be the dosage given in mg at time t.
According to the problem, each hour the drug is 20% less effective than the previous hour. This means that after each hour, the remaining amount of medication is 80% (or 0.8) of the previous amount.
Based on this information, we can write the equation as follows:
R(t) = D(t) * (0.8)^t
In this equation, D(t) represents the dosage given at time t, and (0.8)^t represents the decrease in effectiveness of the drug over time.
For the given problem, the initial dose is 200 mg, so the equation becomes:
R(t) = 200 * (0.8)^t
This equation shows the relationship between the amount of medication remaining in the patient's bloodstream and the time in hours since the medication was administered.