Chapter 7.4, Problem 96E

a.

To determine

To find: The equation of parabola

Answer to Problem 96E

The equation of parabola is $y=138.8t^2-26.6t-56.7$

Explanation of Solution

Given information:

The table shows the average annual salaries $y$ (in thousands of dollars) for public school classroom teachers in United States from $2011$ through $2013$ . Let $t$ represent the year, with $t=1$ corresponding to $2011$ .

  $\begin{array}{cc}\text{Year}& \text{Annual salary,y}\\ 2011& 55.5\\ 2012& 55.4\\ 2013& 56.4\end{array}$

Calculation:

Given the equation of parabola: $y=a{t}^2+bt+c$

Putting $t=1$ and $y=55.5$ in $y=a{t}^2+bt+c$

  $a+b+c=55.5$

Putting $t=2$ and $y=55.4$ in $y=a{t}^2+bt+c$

  $4a+2b+c=55.4$

Putting $t=3$ and $y=56.4$ in $y=a{t}^2+bt+c$

  $9a+3b+c=56.4$

Equation 1: $a+b+c=55.5$

Equation 2: $4a+2b+c=55.4$

Equation 3: $9a+3b+c=56.4$

Now solving this system of equation

Augmented matrix: $\left(\begin{array}{ccccc}1& 1& 1& ⋮& 55.5\\ 4& 2& 1& ⋮& 55.4\\ 9& 3& 1& ⋮& 56.4\end{array}\right)$

Reducing this augmented matrix in echelon form

  $\begin{array}{l}\left(\begin{array}{ccccc}1& 1& 1& ⋮& 55.5\\ 4& 2& 1& ⋮& 55.4\\ 9& 3& 1& ⋮& 56.4\end{array}\right)\\ R_2\to R_2-4R_1,R_3\to R_3-9R_1\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 55.5\\ 0& -2& -3& ⋮& -166.6\\ 0& -6& -8& ⋮& -4431\end{array}\right)\\ R_2\to \frac{-1}{2}{R}_2\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 55.5\\ 0& 1& \frac{3}{2}& ⋮& \frac{166.6}{2}\\ 0& -6& -8& ⋮& -443.1\end{array}\right)\\ R_3\to R_3+6R_2\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 55.5\\ 0& 1& \frac{3}{2}& ⋮& \frac{166.6}{2}\\ 0& 0& 1& ⋮& -56.7\end{array}\right)\end{array}$

Now system of linear equation according to echelon matrix

  $\begin{array}{l}a+b+c=55.5\\ b+\frac{3c}{2}=\frac{166.6}{2}\\ c=-56.7\\ b=-26.6\\ a=138.8\end{array}$

Hence the equation of parabola is

  $y=138.8t^2-26.6t-56.7$

b.

To determine

To Graph: use a graphing utility to graph the parabola and plot data points

Explanation of Solution

Given information:

The table shows the average annual salaries $y$ (in thousands of dollars) for public school classroom teachers in United States from $2011$ through $2013$ . Let $t$ represent the year, with $t=1$ corresponding to $2011$ .

  $\begin{array}{cc}\text{Year}& \text{Annual salary,y}\\ 2011& 55.5\\ 2012& 55.4\\ 2013& 56.4\end{array}$

Graph:

The equation of parabola is $y=138.8t^2-26.6t-56.7$

  

c.

To determine

To Find: average annual salaries in $2015$ , $2020$ and $2025$

Answer to Problem 96E

Average annual salaries in $2015$ , $2020$ and $2025$ is $3280.3$ , $13557.3$ and $30774.3$ thousands dollars

Explanation of Solution

Given information:

The table shows the average annual salaries $y$ (in thousands of dollars) for public school classroom teachers in United States from $2011$ through $2013$ . Let $t$ represent the year, with $t=1$ corresponding to $2011$ .

  $\begin{array}{cc}\text{Year}& \text{Annual salary,y}\\ 2011& 55.5\\ 2012& 55.4\\ 2013& 56.4\end{array}$

Calculation:

The equation of parabola is $y=138.8t^2-26.6t-56.7$

Year $2015$ corresponds to $t=5$

Average annual salaries in $2015$ :

  $\begin{array}{l}{y}_1=138.8{\left(5\right)}^2-26.6\left(5\right)-56.7\\ y_1=3280.3\end{array}$

Year $2020$ corresponds to $t=10$

Average Consumer costs in $2020$ :

  $\begin{array}{l}{y}_2=138.8{\left(10\right)}^2-26.6\left(10\right)-56.7\\ y_2=13557.3\end{array}$

Year $2025$ corresponds to $t=15$

Average Consumer costs in $2025$ :

  $\begin{array}{l}{y}_3=138.8{\left(15\right)}^2-26.6\left(15\right)-56.7\\ y_3=30774.3\end{array}$

Hence, average consumer costs in $2015$ , $2020$ and $2025$ is $3280.3$ , $13557.3$ and $30774.3$ thousands dollars

d.

To determine

To Explain: Are estimates in part (c) reasonable?

Answer to Problem 96E

Yes

Explanation of Solution

Given information:

The table shows the average annual salaries $y$ (in thousands of dollars) for public school classroom teachers in United States from $2011$ through $2013$ . Let $t$ represent the year, with $t=1$ corresponding to $2011$ .

  $\begin{array}{cc}\text{Year}& \text{Annual salary,y}\\ 2011& 55.5\\ 2012& 55.4\\ 2013& 56.4\end{array}$

Yes, estimates in part (c) is reasonable because it can be seen from given table that as the time increases average annual salaries increases also in $2015$ , $2020$ and $2025$ average annual salaries is $3280.3$ , $13557.3$ and $30774.3$ thousands dollars which is also increasing.