solving a mixing tank model?

Question

the inital concentration of a 1000liter tank is given Co, find C(t) if the feedback control law is
s(c)= 0, if c>k
z, if c< or = k
show that t* is a decreasing function of C,and therefore to minimize t*, take z as large as possible. Assume Volume=500liters. k=optimal value,c(t)=concentration

Answer 1

c(t) = [1 - exp(-z * (t+to)] for 0 <= t <= t*
c(to) = Co = [1 - exp(-z * to)]
c(t*) = k
c(t) = k for t >= t*
Co and k are in [0,1]

(t* + to) is minimized by maximizing z

1-Co = exp(-z*to)
ln(1-Co) = -z*to
to = (-1/z) * ln(1-Co)

k = 1 - exp(-z * (t*+to))
1-k = exp(-z * (t*+to))
ln(1-k) = -z * (t*+to)
t* = (1/z) * [ln(1-Co) - ln(1-k)]
= (1/z)*ln((1-Co) / (1-k))
As Co approaches k from below, t* decreases to zero
also, as z goes up from zero, t* decreases

Don't know how to use the V=500 liters data.