Patients with acute or serious infections, such as Salmonella typhi or gram-negative bacteria causing meningitis, can be successfully treated with regular doses of Chloramphenicol. A patient weighing 150 pounds requires a dose of 850 milligrams, while a patient weighing 300 lbs requires a dose of 1700 milligrams.
Lab experiments have indicated that the concentration level C, in micrograms per milliliter (mcg/ml), of the medication in the a patient’s bloodstream as a function of the amount of time t, in hours, since the medication is administered is given by the quadratic function:
C(t)= -1.152t^2 + 8.064t + 4.288. For a drug to have a beneficial effect, its concentration in the bloodstream must exceed a certain value, called the minimum therapeutic level. The minimum therapeutic level of Chloramphenicol is 8 mcg/ml.
Determine a linear function for determining the dosage needed for a patient based on the patient’s weight, in pounds.
To determine a linear function for determining the dosage needed for a patient based on their weight, we can start by examining the given information. We know that a patient weighing 150 pounds requires a dose of 850 milligrams and a patient weighing 300 pounds requires a dose of 1700 milligrams.
Let's denote the patient's weight as W (in pounds) and the dosage needed as D (in milligrams). We can form an equation using these values:
D = mW + b
We need to find the values of m and b to form the linear function.
Using the given information, we can substitute the weight and dosage values for the two patients into the equation to get two equations:
For the patient weighing 150 pounds: 850 = m(150) + b
For the patient weighing 300 pounds: 1700 = m(300) + b
Now we have a system of linear equations with two variables (m and b). We can solve this system to find the values of m and b.
Let's solve the system of equations:
850 = m(150) + b (Equation 1)
1700 = m(300) + b (Equation 2)
First, we can rearrange Equation 1 to isolate b:
b = 850 - m(150)
Next, substitute this expression for b in Equation 2:
1700 = m(300) + 850 - m(150)
Now simplify the equation:
1700 = 300m + 850 - 150m
Combine like terms:
1700 = 150m + 850
Subtract 850 from both sides:
850 = 150m
Divide both sides by 150:
m = 850 / 150
m = 5.67
Now that we have the value of m, we can substitute it back into Equation 1 to find the value of b:
850 = 5.67(150) + b
Simplify the equation:
850 = 850.5 + b
Subtract 850.5 from both sides:
b = 850 - 850.5
b ≈ -0.5
Therefore, the linear function for determining the dosage needed for a patient based on their weight is:
D = 5.67W - 0.5, where D is the dosage needed in milligrams, and W is the patient's weight in pounds.