Answer:
0.4865
Explanation:
The reaction rate to a new drug t hours after the drug is administered is given by the expression r'(t)=0.5te^-t.
First, we find an expression for r(t), which is the total reaction to the drug, stating from t=0, when the drug was administered by integrating r'(t).
Here, we employ the "{" sign to represent integration, and apply the method of integrating by parts as follows: {udv = uv - {vdu
r = {0.5te^-tdt
To do this we let u = 0.5t, and dv = (e^-t) dt
Thus du = 0.5 dt and v = - e^-t
Therefore r = 0.5 t * (- e^-t) - {0.5*- e^-t dt =
= - 0.5 t * ( e^-t) -0.5* e^-t + C
To evaluate the C, we plug in r =0 at t=0
0 = 0-0.5+C
c = 0.5
Thus r(t) = 0.5- 0.5 t * ( e^-t) -0.5* e^-t = 0.5 -(0.5e^-t)( t -1 )
Over the first 5 hours we obtain r(5) as follows:
r(5) = 0.5 - 0.5*5 * ( e^-5) -0.5* e^--5 = 0.4865