   Chapter 11.2, Problem 72E

Chapter
Section
Textbook Problem

# After injection of a dose D of insulin, the concentration of insulin in a patient’s system decays exponentially and so it can be written as De−at, where t represents time in hours and a is a positive constant. (a) If a dose D is injected every T hours, write an expression for the sum of the residual concentrations just before the (n + 1)st injection. (b) Determine the limiting pre-injection concentration. (c) If the concentration of insulin must always remain at or above a critical value C, determine a minimal dosage D in terms of C, a, and T.

(a)

To determine

To find: The sum of the residual concentrations just before the (n+1)st injection.

Explanation

Given:

The concentration of insulin in a patient’s system is Deat.

Result used:

The sum of the finite geometric series (k=1nark1 (or) a+ar+ar2++arn1) is a(1rn)1r, where a is the first term and r is the common ratio of the series.

Calculation:

The residual concentration just before the second injection is DeaT.

The residual concentration just before the third injection is Dea2T.

The residual concentration just before the fourth injection is Dea3T.

Proceed in the similar way, the residual concentrations just before the (n+1)st injection is DeanT.

The sum of the residual concentrations just before the (n+1)st injection Cn is computed as follows,

Cn=DeaT+Dea2T+Dea3T++Dea(n1)T+DeanT=D(eaT)1+D(eaT)2+D(eaT)3++D(eaT

(b)

To determine

The limiting pre-injection concentration.

(c)

To determine

The minimal dosage D in terms of C, a and T.

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