Chapter 12.3, Problem 44E

a.

To determine

To find the fraction of dose left in the blood stream before the second dose is administered.

Answer to Problem 44E

  $\frac{1}{8}D$

Explanation of Solution

Given:

The half-life of certain drug in bloodstream is $2$ hours.

Calculation:

After every $2$ hours the fraction remains half of the previous one.

After $2$ hours, $\frac{1}{2}D$ remains.

After $4$ hours, $\frac{1}{2}\left(\frac{1}{2}D\right)=\frac{1}{4}D$ remains.

After $6$ hours, $\frac{1}{2}\left(\frac{1}{4}D\right)=\frac{1}{8}D$ remains.

b.

To determine

To write the general expression for the geometric series for $nth$ dose.

Answer to Problem 44E

  $a_n=D{\left(\frac{1}{8}\right)}^{n-1}$

Explanation of Solution

Given:

The half-life of certain drug in bloodstream is $2$ hours.

Calculation:

Each dose occurs after every hours, and thus we have,

The remaining $=\frac{1}{8}\times$ the remaining of the previous does.

The situation describes the geometric series for $a_1=D,r=\frac{1}{8}$

Therefore, the general expression for the geometric series for $nth$ dose is

  $a_n=D{\left(\frac{1}{8}\right)}^{n-1}$

c.

To determine

To find the amount of medicine present in the bloodstream for the large values of $n$ .

Answer to Problem 44E

  $\frac{8}{7}D$

Explanation of Solution

Given:

The half-life of certain drug in bloodstream is $2$ hours.

Calculation:

For large values of $n$ , $n\to \infty$

Sum of $n$ terms of geometric sequence is given by

  $\begin{array}{l}{S}_n=\frac{a}{1-r}\\ =\frac{D}{1-\frac{1}{8}}=\frac{D}{\frac{7}{8}}\\ =\frac{8}{7}D\end{array}$