The quantity Q, in milligrams, of medication in a patient is modelled by the formula Q=238*(0.8)^(t/4), where t is the number of hours since the initial dose.

Question

(a) What was the initial dose of the medication?
(b) By what percentage is Q decreasing every 4 hours?
(c) By what percentage is Q decreasing every 2 hours?
(d) Calculate ∆Q/∆t from t = 2 to t = 2+1/6. What are the units of this number? What does this number tell you?
(e) Without further calculation, is the number from the previous part a good estimate of Q'(2)?

Answer 1

(a) The initial dose of the medication is Q(0) = 238*(0.8)^(0/4) = 238 milligrams.

(b) Q is decreasing every 4 hours by a factor of 0.8, which is a 20% decrease. This can be seen by noticing that Q(t+4) = 238*(0.8)^((t+4)/4) = 238*(0.8)^(t/4)*0.8 = Q(t)*0.8.

(c) Q is decreasing every 2 hours by a factor of (0.8)^(1/2), which is a 10% decrease. This can be seen by noticing that Q(t+2) = 238*(0.8)^((t+2)/4) = 238*(0.8)^(t/4)*(0.8)^(1/2) = Q(t)*(0.8)^(1/2).

(d) ∆Q/∆t from t=2 to t=2+1/6 is [(Q(2+1/6)-Q(2))/((2+1/6)-2)]/1000. Evaluating Q(2+1/6) and Q(2) gives ∆Q/∆t ≈ -2.57 milligrams per hour squared. The units of this number are milligrams per hour squared, and it represents the acceleration of the decrease in the medication quantity over the given time interval.

(e) No, the number from part (d) is not a good estimate of Q'(2). Q'(2) represents the instantaneous rate of change of Q at t=2, and cannot be accurately estimated from the average rate of change over a finite time interval (which is what ∆Q/∆t represents).