A drug is administered every 6 hours. The kidneys eliminate 55% of the drug over that period. The initial dose is 210 mg. Repeated dosage is 70 mg
What is the “Difference equation”? Find the first seven terms of the of the solution sequence. Round to two digits each sequence value.
Graph the “Difference equation in (a).
What is the “Maintenance level” for the medication? Round your answer to two digits.
How much of the drug remains in the body after 18 hours? Use the solution formula to the difference equation. (Page 366).
How much of the drug remains in the body after one week? Use the solution formula to the difference equation. (Page 366).
How much should the repeated dosage be to ensure a maintenance level of 40 mg?
To answer these questions related to the difference equation, we need to understand what the given information implies and then proceed step by step to find the solutions.
1. Difference equation:
The difference equation is a formula that represents the relationship between consecutive terms in a sequence. In this case, the difference equation represents the amount of drug remaining in the body after each dosage.
Let's break down the given information:
- The drug is administered every 6 hours.
- The kidneys eliminate 55% of the drug over that 6-hour period.
- The initial dose is 210 mg.
- The repeated dosage is 70 mg.
From this information, we can conclude that after each dosage, 55% of the drug is eliminated, while the remaining 45% remains in the body.
Therefore, the difference equation for this scenario can be written as:
D(n) = 0.45 * [D(n-1) - 70]
Where:
D(n) represents the amount of drug (in mg) in the body after the nth dosage.
D(n-1) represents the amount of drug remaining in the body before the current dosage.
2. First seven terms of the solution sequence:
To find the first seven terms of the solution sequence, we'll start with the initial dose of 210 mg and apply the difference equation repeatedly.
D(0) = 210 (initial dose)
Using the difference equation, we can find the subsequent terms:
D(1) = 0.45 * [D(0) - 70]
D(2) = 0.45 * [D(1) - 70]
...
D(6) = 0.45 * [D(5) - 70]
Calculating each term using the given values will give us the first seven terms of the solution sequence, rounded to two digits.
3. Graphing the difference equation:
To graph the difference equation, we can plot the values of the solution sequence (amount of drug remaining after each dosage) on the y-axis against the corresponding dosage number on the x-axis.
4. Maintenance level:
The maintenance level for the medication is the amount of drug remaining in the body after the effects of elimination have stabilized. It is the level at which the dosage compensates for the elimination rate.
To find the maintenance level in this scenario, we need to determine the value of D(n) when it becomes constant. This can be achieved by repeatedly applying the difference equation until the solution sequence stabilizes. The maintenance level is the constant value obtained.
5. Drug remaining after 18 hours:
To calculate the amount of drug remaining in the body after 18 hours, we can use the solution formula to the difference equation. This formula allows us to directly compute D(n) for any given n.
Similarly, to find the drug remaining after one week, we can use the solution formula and substitute the appropriate values.
6. Dosage for a maintenance level of 40 mg:
To ensure a maintenance level of 40 mg, we need to find the repeated dosage value that will compensate for the elimination rate and leave the desired amount in the body. We can use the difference equation and solve for the repeated dosage amount.