Chapter 7.4, Problem 95E

a.

To determine

To Find: The equation of parabola

Answer to Problem 95E

The equation of parabola is $y=24t^2+67t+1831$

Explanation of Solution

Given information:

The table shows the average annual consumer costs $y$ (in dollars) for health insurance from $2010$ to $2012$ . Let $t$ represent the year, with $t=0$ corresponding to $2010$ .

  $\begin{array}{cc}\text{Year}& \text{Cost},y\\ 2010& 1831\\ 2011& 1922\\ 2012& 2061\end{array}$

Calculation:

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

Putting $t=0$ and $y=1831$ in $y=a{t}^2+bt+c$

  $c=1831$

Putting $t=1$ and $y=1922$ in $y=a{t}^2+bt+c$

  $a+b+c=1922$

Putting $t=2$ and $y=2061$ in $y=a{t}^2+bt+c$

  $4a+2b+c=2061$

Equation 1: $a+b+c=1922$

Equation 2: $4a+2b+c=2061$

Equation 3: $c=1831$

Now solving this system of equation

Augmented matrix: $\left(\begin{array}{ccccc}1& 1& 1& ⋮& 1922\\ 4& 2& 1& ⋮& 2061\\ 0& 0& 1& ⋮& 1831\end{array}\right)$

Reducing this augmented matrix in echelon form

  $\begin{array}{l}\left(\begin{array}{ccccc}1& 1& 1& ⋮& 1922\\ 4& 2& 1& ⋮& 2061\\ 0& 0& 1& ⋮& 1831\end{array}\right)\\ R_2\to R_2-4R_1\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 1922\\ 0& -2& -3& ⋮& -5627\\ 0& 0& 1& ⋮& 1831\end{array}\right)\\ R_2\to \frac{-1}{2}{R}_2\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 1922\\ 0& 1& \frac{3}{2}& ⋮& \frac{5627}{2}\\ 0& 0& 1& ⋮& 1831\end{array}\right)\end{array}$

Now system of linear equation according to echelon matrix

  $\begin{array}{l}a+b+c=1922\\ b+\frac{3c}{2}=\frac{5627}{2}\\ c=1831\\ b=67\\ a=24\end{array}$

Hence the equation of parabola is

  $y=24t^2+67t+1831$

b.

To determine

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

Explanation of Solution

Given information:

The table shows the average annual consumer costs $y$ (in dollars) for health insurance from $2010$ to $2012$ . Let $t$ represent the year, with $t=0$ corresponding to $2010$ .

  $\begin{array}{cc}\text{Year}& \text{Cost},y\\ 2010& 1831\\ 2011& 1922\\ 2012& 2061\end{array}$

Graph:

The equation parabola is $y=24t^2+67t+1831$

  

c.

To determine

To Find: average consumer costs in $2015$ , $2020$ and $2025$

Answer to Problem 95E

Average consumer costs in $2015$ , $2020$ and $2025$ is $2766\$$ , $4901\$$ and $8236\$$

Explanation of Solution

Given information:

The table shows the average annual consumer costs $y$ (in dollars) for health insurance from $2010$ to $2012$ . Let $t$ represent the year, with $t=0$ corresponding to $2010$ .

  $\begin{array}{cc}\text{Year}& \text{Cost},y\\ 2010& 1831\\ 2011& 1922\\ 2012& 2061\end{array}$

Calculation:

The equation of parabola is $y=24t^2+67t+1831$

Year $2015$ corresponds to $t=5$

Average Consumer costs in $2015$ :

  $\begin{array}{l}{y}_1=24{\left(5\right)}^2+67\left(5\right)+1831\\ y_1=2766\$\end{array}$

Year $2020$ corresponds to $t=10$

Average Consumer costs in $2020$ :

  $\begin{array}{l}{y}_2=24{\left(10\right)}^2+67\left(10\right)+1831\\ y_2=4901\$\end{array}$

Year $2025$ corresponds to $t=15$

Average Consumer costs in $2025$ :

  $\begin{array}{l}{y}_3=24{\left(15\right)}^2+67\left(15\right)+1831\\ y_3=8236\$\end{array}$

Hence, average consumer costs in $2015$ , $2020$ and $2025$ is $2766\$$ , $4901\$$ and $8236\$$

d.

To determine

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

Answer to Problem 95E

Yes

Explanation of Solution

Given information:

The table shows the average annual consumer costs $y$ (in dollars) for health insurance from $2010$ to $2012$ . Let $t$ represent the year, with $t=0$ corresponding to $2010$ .

  $\begin{array}{cc}\text{Year}& \text{Cost},y\\ 2010& 1831\\ 2011& 1922\\ 2012& 2061\end{array}$

Yes,estimates in part (c) is reasonable because it can be seen from given table that as the time increases average consumer cost increases also in $2015$ , $2020$ and $2025$ average consumer cost is $2766\$$ , $4901\$$ and $8236\$$ which is also increasing.