The quantity Q, in milligrams, of medication in a patient is modelled by the formula Q = 238(0.8)^(t/4) where t is in the number of hours since the initial dose. Calculate ∆Q/∆t from t = 2 to t = 2+1/6. What are the units of this number? What does this number tell you?
We start by finding the derivative of Q with respect to t:
dQ/dt = (238/4)(0.8)^(t/4-1) * (1/4)
Simplifying this expression:
dQ/dt = 14.875(0.8)^(t/4-1)
Next, we evaluate ∆Q/∆t from t = 2 to t = 2+1/6:
∆Q/∆t = Q(2+1/6) - Q(2) / (2+1/6 - 2)
Using the given formula, we can find Q(2+1/6) and Q(2):
Q(2+1/6) = 238(0.8)^((2+1/6)/4) = 188.28 mg
Q(2) = 238(0.8)^(2/4) = 149.44 mg
Substituting these values into the formula for ∆Q/∆t:
∆Q/∆t = (188.28 - 149.44) / (1/6) = 236.64 mg/hour
The units of this number are mg/hour. This number tells us the rate at which the medication concentration in the patient's body is changing over time from hour 2 to hour 2+1/6. Specifically, it tells us that the medication concentration is increasing at a rate of 236.64 mg per hour over this time period.