Chapter 7.4, Problem 89E

a.

To determine

To find: The quadratic function whose graph passes through the given points.

Answer to Problem 89E

The quadratic function whose graph passes through the given points is , $y=-\frac{1}{250}{x}^2+\frac{11}{30}x+5.$

Explanation of Solution

Given information: The table shows the horizontal distances x and vertical distances y

Of a thrown baseball at three specific times, where x and y are measured in feet.

Horizontal distance, x	Height y
0	5.0
15	9.6
30	12.4

Calculation:

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

Using the data given, the system of equations is:

  $\begin{array}{c}c=5\\ 225a+15b+c=9.6\\ 900a+30b+c=12.4\end{array}$

Now solving this system of equation

Augmented matrix: $\left[\begin{array}{ccc}0& 0& 1\\ 225& 15& 1\\ 900& 30& 1\end{array}\begin{array}{c}5\\ 9.6\\ 12.4\end{array}\right]$ .

Reducing this augmented matrix in echelon form.

Get a leading 1 in row 1:

  $\frac{1}{225}{R}_2\leftrightarrow R_1\left[\begin{array}{ccc}1& \frac{1}{15}& \frac{1}{225}\\ 0& 0& 1\\ 900& 30& 1\end{array}\begin{array}{c}\frac{16}{375}\\ 5\\ 12.4\end{array}\right]$

Get 0s below the leading 1 in row 1:

  $R_3-900R_1\to R_3\left[\begin{array}{ccc}1& \frac{1}{15}& \frac{1}{225}\\ 0& 0& 1\\ 0& -30& -3\end{array}\begin{array}{c}\frac{16}{375}\\ 5\\ -26\end{array}\right]$

Get a leading 1 in row 2:

  $-\frac{1}{30}{R}_3\leftrightarrow R_2\left[\begin{array}{ccc}1& \frac{1}{15}& \frac{1}{225}\\ 0& 1& \frac{1}{10}\\ 0& 0& 1\end{array}\begin{array}{c}\frac{16}{375}\\ \frac{13}{15}\\ 5\end{array}\right]$

Write the system of linear equations corresponding to the reduced row-echelon form of the matrix:

  $\begin{array}{c}a+\frac{1}{15}b+\frac{1}{225}c=\frac{16}{375}.\\ b+\frac{1}{10}c=\frac{13}{15}.\\ c=5.\end{array}$

Use back-substitution to find a and b .

  $b=\frac{11}{30},a=-\frac{1}{250}$ .

The equation of the parabola is:

  $y=-\frac{1}{250}{x}^2+\frac{11}{30}x+5.$

b.

To determine

To graph: The quadratic function using the graphing utility and approximate the maximum height of the ball using the graph.

Answer to Problem 89E

Maximum height $\approx 13.4ft$ .

Explanation of Solution

Given information: The table shows the horizontal distances x and vertical distances y

Of a thrown baseball at three specific times, where x and y are measured in feet.

Horizontal distance, x	Height y
0	5.0
15	9.6
30	12.4

Calculation:

Using a graphing utility the graph of the function $y=-\frac{1}{250}{x}^2+\frac{11}{30}x+5.$ is shown below.

  

From the graph,

Maximum height $\approx 13.4ft$ .