Chapter 1.4, Problem 69E

To determine

To find the percent of prescriptions filled with generic drugs in each year from $2004$ through $2010$ .

Answer to Problem 69E

  $\begin{array}{cccccccc}Year\left(t\right)& 2004& 2005& 2006& 2007& 2008& 2009& 2010\\ \begin{array}{l}\%age\\ of\\ prescription\left(p\right)\end{array}& 45.58\%& 50.15\%& 54.72\%& 59.29\%& 64.4\%& 67.75\%& 71.10\%\end{array}$

Explanation of Solution

Given information:

The percents $p$ of prescriptions filled with generic drugs in the United States from $2004$ through $2010$ (see figure) can be approximated by the model.

  $p\left(t\right)=\{\begin{array}{l}4.57t+27.3,\\ 3.35t+37.6,\end{array}$$\begin{array}{l}4\le t\le 7\\ 8\le t\le 10\end{array}$

Where $t$ represents the year, with $t=4$ corresponding to $2004$ . Use this model to find the percent of prescriptions filled with generic drugs in each year from $2004$ through $2010$ .

  

Calculation:

The percents $p$ of prescriptions filled with generic drugs from $2004$ through $2010$ can be approximated by the model.

  $p\left(t\right)=\{\begin{array}{l}4.57t+27.3,\\ 3.35t+37.6,\end{array}$$\begin{array}{l}4\le t\le 7\\ 8\le t\le 10\end{array}$

From year $2004-2007$ use $p\left(t\right)=4.57t+27.3$ and from year $2008-2010$

  $p\left(t\right)=3.35+37.6$

Now find percents $p$ of prescriptions filled with generic drugs from $2004-2007$

Put $t=4,5,6,7$ in $p\left(t\right)=4.57t+27.3$ we get,

  $\begin{array}{l}p\left(4\right)=4.57\left(4\right)+27.3\\ =18.28+27.3\\ =45.58\%\\ p\left(5\right)=4.57\left(5\right)+27.3\\ =22.85+27.3\\ =50.15\%\\ p\left(6\right)=4.57\left(6\right)+27.3\\ =27.42+27.3\\ =54.72\%\\ p\left(7\right)=4.57\left(7\right)+27.3\\ =31.99+27.3\\ =59.29\%\end{array}$

Now find percents $p$ of prescriptions filled with generic drugs from $2008-2010$

Put $t=8,9,10$ in $p\left(t\right)=3.35t+37.6$ we get,

  $\begin{array}{l}p\left(8\right)=3.35\left(8\right)+37.6\\ =26.8+37.6\\ =64.4\%\\ p\left(9\right)=3.35\left(9\right)+37.6\\ =30.15+37.6\\ =67.75\%\\ p\left(10\right)=3.35\left(10\right)+37.6\\ =33.5+37.6\\ =71.1\%\end{array}$

Hence, the percents $p$ of prescriptions filled with generic drugs from $2004$ through $2010$ is

Tabulate as,

  $\begin{array}{cccccccc}Year\left(t\right)& 2004& 2005& 2006& 2007& 2008& 2009& 2010\\ \begin{array}{l}\%age\\ of\\ prescription\left(p\right)\end{array}& 45.58\%& 50.15\%& 54.72\%& 59.29\%& 64.4\%& 67.75\%& 71.10\%\end{array}$