In diabetic people, the regulatory action of insulin is not sufficient to prevent the buildup of glucose in the blood stream. Several models have been developed to predict the the glucose$-$insulin interdependence, upon which diagnostic tests were based. In one of the proposed

models, the blood stream is modeled as an infinite number of equal volume continuous stirred tanks in series, as shown below. The blood flow rate is Q and the volume of each tank is $V.$

Assume initially there is no glucose in the blood stream, and at time $t=0$ glucose is injected rapidly into the blood $($i$.$e$.$ into Tank $1).$ Show that for Tank $\text{'}i\text{'}$:

$\frac{\frac{?}{\text{b}\text{a}\text{r}}(C{)}_i}{\frac{?}{\text{b}\text{a}\text{r}}(C{)}_{i-1}}=\frac{1}{s+1}$

here $\frac{?}{\text{b}\text{a}\text{r}}(C{)}_i$ is the Laplace transform of $C_i,$ and $=\frac{V}{Q}$ is the residence time of the tank $\text{'}i\text{'}.$

If the amount of glucose that is rapidly injected into Tank $1$ is equal to $N_0,$ find an expression for $C_N(t)$ in terms of $,V,Q,$ and $N_o$