Answer:
Explanation:
1. The function f(t) represents the concentration of medication in the bloodstream over time. The function has a parabolic shape, with the maximum value at t = 8 hours.
To find the maximum concentration of the medication, we can set the derivative of the function equal to 0 and solve for t. The derivative of f(t) is:
f'(t) = -30t + 120
Setting this equal to 0 and solving for t, we get:
-30t + 120 = 0
-30t = -120
t = 4
Therefore, the maximum concentration of the medication occurs at t = 4 hours. At this time, the concentration of the medication is:
f(4) = -15(4)^2 + 120(4) = 120 mg
It's important to note that this function represents the concentration of the medication in the bloodstream, not the total amount of medication that has been taken. The total amount of medication taken over a given time period can be found by integrating the function over that time period.
2. To determine the maximum amount of medication in the bloodstream, we need to find the maximum concentration of the medication and then multiply that concentration by the volume of the bloodstream.
We have already determined that the maximum concentration of the medication is 120 mg/L. The volume of the bloodstream in an adult is approximately 5 liters. Therefore, the maximum amount of medication in the bloodstream is:
120 mg/L * 5 L = 600 mg
This is the maximum amount of medication that can be present in the bloodstream at any given time. It's important to note that this amount may be affected by various factors, including the rate at which the medication is metabolized and eliminated from the body.
3. To determine the concentration of medication in the bloodstream after 2 hours, we can substitute 2 for t in the function f(t) and solve for the concentration. The function for the concentration of medication in the bloodstream is:
f(t) = -15t^2 + 120t
Substituting 2 for t, we get:
f(2) = -15(2)^2 + 120(2) = 60 mg/L
Therefore, the concentration of the medication in the bloodstream after 2 hours is 60 mg/L. It's important to note that this represents the concentration of the medication at a specific point in time, not the total amount of medication that has been taken. The total amount of medication taken over a given time period can be found by integrating the function over that time period.