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.
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.
How much of the drug remains in the body after one week? Use the solution formula to the difference equation.
How much should the repeated dosage be to ensure a maintenance level of 40 mg?
To answer these questions, we need to understand the concept of a difference equation. A difference equation is a mathematical equation used to model the change in a variable over discrete time intervals. In this case, we want to model the amount of the drug remaining in the body after each dose.
Let's denote the sequence of drug amounts as {Dn}, where Dn represents the amount of drug after the nth dose. To find the difference equation, we need to determine how the amount of drug changes from one dose to the next.
1. Difference Equation:
The initial dose is 210 mg, and the kidneys eliminate 55% of the drug over each 6-hour period. Therefore, the amount eliminated per dose is 0.55 * 210 mg = 115.5 mg. The remaining amount after each dose is the previous dose minus the amount eliminated, which can be expressed as:
Dn = Dn-1 - 115.5, where n > 1
For the first dose (n = 1), D1 = 210 mg.
2. First Seven Terms of the Solution Sequence:
Using the difference equation, we can calculate the values for the first seven terms of the sequence {Dn}:
D1 = 210 mg (initial dose)
D2 = D1 - 115.5 mg
D3 = D2 - 115.5 mg
D4 = D3 - 115.5 mg
D5 = D4 - 115.5 mg
D6 = D5 - 115.5 mg
D7 = D6 - 115.5 mg
Rounding each term to two digits, we find the first seven terms of the sequence.
3. Maintenance Level:
The maintenance level is the amount of the drug that remains in the body after continuous dosing without any increase or decrease. It occurs when the amount eliminated equals the amount administered per dose. In this case, the maintenance level occurs when the amount eliminated (115.5 mg) equals the repeated dosage (70 mg).
So, the maintenance level is 70 mg.
4. Amount of Drug After 18 Hours:
To find the amount of the drug remaining in the body after 18 hours, we can use the solution formula to the difference equation:
Dn = D1 - (n-1) * 115.5, where n denotes the number of 6-hour intervals passed.
Since 18 hours is equivalent to 3 * 6-hour intervals, we can substitute n = 3 into the formula:
D3 = D1 - (3-1) * 115.5
Substituting D1 = 210 mg and calculating the result, we find the amount of the drug remaining after 18 hours.
5. Amount of Drug After One Week:
Similar to the previous question, we can use the solution formula to find the amount of the drug remaining after one week (168 hours):
Dn = D1 - (n-1) * 115.5, where n denotes the number of 6-hour intervals passed.
Since one week is equivalent to 168 / 6 = 28 intervals, we can substitute n = 28 into the formula:
D28 = D1 - (28-1) * 115.5
Substituting D1 = 210 mg and calculating the result, we find the amount of the drug remaining after one week.
6. Repeated Dosage for a Maintenance Level of 40 mg:
To ensure a maintenance level of 40 mg, the repeated dosage should equal the amount eliminated, which is 115.5 mg.
Therefore, the repeated dosage should be 115.5 mg.