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 find the difference equation, we need to understand the pattern in how the drug is administered and eliminated.
The drug is administered every 6 hours, and the kidneys eliminate 55% of the drug over that period.
Let's break it down step by step to find the difference equation:
1. The initial dose is 210 mg: D₀ = 210 mg (Initial dose)
2. The kidneys eliminate 55% of the drug over the 6-hour period: E = 55% = 0.55 (Elimination rate)
3. The repeated dosage is 70 mg given every 6 hours: R = 70 mg (Repeated dosage)
Now, let's find the difference equation:
At each time interval, the amount of the drug in the body would be equal to the amount from the previous interval minus the elimination rate plus the repeated dosage.
Difference equation: D(n+1) = D(n) - (D(n) * E) + R
The first seven terms of the solution sequence can be found by using the difference equation and the initial dose.
Term 1: D(1) = D₀ - (D₀ * E) + R
Term 2: D(2) = D(1) - (D(1) * E) + R
Term 3: D(3) = D(2) - (D(2) * E) + R
...
Term 7: D(7) = D(6) - (D(6) * E) + R
Now, let's calculate the first seven terms of the solution sequence by substituting the given values into the difference equation and rounding each sequence value to two decimal places.
Term 1: D(1) = 210 - (210 * 0.55) + 70
Term 2: D(2) = D(1) - (D(1) * 0.55) + 70
Term 3: D(3) = D(2) - (D(2) * 0.55) + 70
...
Term 7: D(7) = D(6) - (D(6) * 0.55) + 70
Now, let's move on to the next question.
The maintenance level for the medication refers to the amount of the drug remaining in the body after repeated dosages over time. To find this level, we need to determine the point at which the amount of the drug remaining reaches a stable value.
Using the difference equation, we can set D(n+1) = D(n) and solve for D.
D(n) - (D(n) * E) + R = D(n)
Simplifying, we get:
(D(n) * (1 - E)) = R
D(n) = R / (1 - E)
Substituting the given values, we find:
Maintenance Level = 70 / (1 - 0.55)
Now, let's answer the next question.
To calculate how much of the drug remains in the body after 18 hours, we can use the solution formula to the difference equation:
D(n) = (D₀ * (1 - E)^(n-1)) + (R * ((1 - E^(n-1)) / (1 - E)))
Substituting the given values:
D(3) = (210 * (1 - 0.55)^(3-1)) + (70 * ((1 - 0.55^(3-1)) / (1 - 0.55)))
Now, let's solve for D(3) and round the value to two decimal places.
Finally, let's move on to the last two questions.
To find how much of the drug remains in the body after one week (168 hours), we can use the solution formula:
D(n) = (D₀ * (1 - E)^(n-1)) + (R * ((1 - E^(n-1)) / (1 - E)))
Substituting the given values:
D(28) = (210 * (1 - 0.55)^(28-1)) + (70 * ((1 - 0.55^(28-1)) / (1 - 0.55)))
Now, let's solve for D(28) and round the value to two decimal places.
Lastly, to ensure a maintenance level of 40 mg, we need to find the value of the repeated dosage (R). Using the solution formula for D(n) and substituting the given values, we can solve for R:
40 = (210 * (1 - 0.55)^(n-1)) + (R * ((1 - 0.55^(n-1)) / (1 - 0.55)))
Now, let's solve for R and round the value to two decimal places.