Chapter 6.2, Problem 23PPS

a.

To determine

To find: Variables, equations.

Answer to Problem 23PPS

Variables − Supply(s), Demand(d), and time(t in years)

  $s\left(t\right)=5599.9t+1,890,000$

  $d\left(t\right)=40,520.7t+2,000,000$

Explanation of Solution

Given information:

Demand in $2000=2,000,000$

Supply in $2000=\text{ 1,890,000}$

Demand in $2020=\text{ 2,810,414}$

Supply in $2020=\text{ 2,001,998}$

The variables are Supply(s), Demand(d) and time(t in years).

Consider year $2000$ as time (in years) $t=0$ .

Demand as a function of $t$ as $d\left(t\right)$ . And Supply as function of $t$ as $s\left(t\right)$ .

Let

  $\begin{array}{l}d\left(t\right)=at+b\\ s\left(t\right)=pt+q\end{array}$

From given information,

  $\begin{array}{l}d\left(0\right)=2,000,000\\ d\left(20\right)=2,810,414\end{array}$

  $\begin{array}{l}d\left(0\right)=2,000,000\\ \Rightarrow a(0)+b=2,000,000\\ \Rightarrow b=2,000,000\end{array}$

  $\begin{array}{l}d\left(20\right)=2,810,414\\ \Rightarrow a(20)+b=2,810,414\\ \Rightarrow 20a+2,000,000=2,810,414\\ \Rightarrow 20a=2,810,414-2,000,000\\ \Rightarrow 20a=810,414\\ \Rightarrow a=40,520.7\end{array}$

Therefore, $d\left(t\right)=40,520.7t+2,000,000$

Similarly,

  $\begin{array}{l}s\left(0\right)=1,890,000\\ \Rightarrow p\left(0\right)+q=1,890,000\\ \Rightarrow q=1,890,000\\ \end{array}$

  $\begin{array}{l}s\left(20\right)=2,001,998\\ \Rightarrow p\left(20\right)+q=2,001,998\\ \Rightarrow 20p+1,890,000=2,001,998\\ \Rightarrow 20p=2,001,998-1,890,000\\ \Rightarrow 20p=111,998\\ \Rightarrow p=5599.9\end{array}$

Therefore, $s\left(t\right)=5599.9t+1,890,000$

b.

To determine

To find: The year in which supply of nurses and demand of nurses are equal..

Answer to Problem 23PPS

1996

Explanation of Solution

Given information:

Demand in $2000=2,000,000$

Supply in $2000=\text{ 1,890,000}$

Demand in $2020=\text{ 2,810,414}$

Supply in $2020=\text{ 2,001,998}$

Let $t=x$ ,for which $s\left(t\right)=d\left(t\right)$

  $\begin{array}{l}s\left(x\right)=d\left(x\right)\\ \Rightarrow 5599.9x+1,890,000=40,520.7x+2,000,000\\ \Rightarrow 1,890,000-2,000,000=40,520.7x-5599.9x\\ \Rightarrow -110,000=34,520.8x\\ \Rightarrow x=-3.18\end{array}$

Since the value of $t=x=-3.18$ is negative, the supply equaled the demand before the year $2000$ .

  $3.18$ is between $3$ and $4$ , so Supply equaled Demand in year $1996$ .