Chapter 1.7, Problem 66PS

Solution

a.

To write: The function in vertex form.

The function in vertex form is $y=-0.0110{\left(x-55.91\right)}^2+39.8852$ .

Given:

The path of a ball thrown by a softball player can be modeled by the function $y=-0.0110x^2+1.23x+5.50$ where $x$ is the softball's horizontal position (in feet) and $y$ is the corresponding height (in feet).

Calculation:

Rewrite the function $y=-0.0110x^2+1.23x+5.50$ into vertex form as given below.

  $\begin{array}{c}y=-0.0110x^2+1.23x+5.50\\ y=-0.0110\left(x^2-111.82x\right)+5.50\\ y+\left(-0.0110\right)\left(3125.93\right)=-0.0110\left(x^2-111.82x+3125.93\right)+5.50\\ y=-0.0110{\left(x-55.91\right)}^2+39.8852\end{array}$

b.

To make: A Table of values for the function. Include values of $x$ from $0$ to $120$ in increments of $10$ .

A table is as follows.

  $\begin{array}{cccccccccccccc}x& 0& 10& 20& 30& 40& 50& 60& 70& 80& 90& 100& 110& 120\\ y& 5.50& 16.7& 25.7& 32.5& 37.1& 39.5& 39.7& 37.7& 33.5& 27.1& 18.5& 7.7& -5.3\end{array}$

Given:

The path of a ball thrown by a softball player can be modeled by the function $y=-0.0110x^2+1.23x+5.50$ where $x$ is the softball's horizontal position (in feet) and $y$ is the corresponding height (in feet).

Calculation:

Table is as follows.

  $\begin{array}{cccccccccccccc}x& 0& 10& 20& 30& 40& 50& 60& 70& 80& 90& 100& 110& 120\\ y& 5.50& 16.7& 25.7& 32.5& 37.1& 39.5& 39.7& 37.7& 33.5& 27.1& 18.5& 7.7& -5.3\end{array}$

c.

To draw: A graph and to find the maximum height of the softball.

The maximum height of the softball is $39.8852\text{ feet}$ and the softball travels $55.91\text{ feet}$ .

Given:

The path of a ball thrown by a softball player can be modeled by the function $y=-0.0110x^2+1.23x+5.50$ where $x$ is the softball's horizontal position (in feet) and $y$ is the corresponding height (in feet).

Calculation:

Points can be plotted as follows.

  

Because the vertex form of the given model is $y=-0.0110{\left(x-55.91\right)}^2+39.8852$ , so the vertex is $\left(55.91,39.8852\right)$ .

The maximum height of the softball is the $y-$ coordinate of the vertex $\left(55.91,39.8852\right)$ of the parabola with the vertex form of the given model $y=-0.0110{\left(x-55.91\right)}^2+39.8852$ .

Hence, the maximum height of the softball is $39.8852\text{ feet}$

To find how far the softball travels we have to twice the value of the $x-$ coordinate of the vertex $\left(55.91,39.8852\right)$ of the parabola with the vertex form of the given model

  $y=-0.0110{\left(x-55.91\right)}^2+39.8852$

Hence, the softball travels $55.91\text{ feet}$