Question

A 0.2 ml dose of a drug is injected into a patient steadily for 0.75 seconds. At the end of this time, the quantity, Q, of the drug in the body starts to decay exponentially at a continuous rate of 0.2 percent per second. Using formulas, express Q as a continuous function of time, t, in seconds

Q(t) = ______ if 0 is less than or equal to t less than or equal to ______
and Q(t) = ______ if _____ less than or equal to t less than or equal to Infinity

Answer 1

Q(t+1 second ) = Q(t) -.002Q(t)
so
dQ/dt = -.002 Q
dQ/Q = -.002 dt

lnQ = -.002 t + c'

Q = e^-.002 t + e^c'
Q = e^c^ e^-.002 t
or
Q = C e^-.002 t in ml
where C is Q at t = 0(start of decay period)
now
first period, t = 0 to t = .75
rate = .2ml/.75 s = .267 ml/s
so
Q(t) = _.267 * t ml for 0<t<.75

and second period
Q(t) = .2 e^-.002(t-.75) ml
for .75<t<oo