91Ó°ÊÓ

The given table is,

We need to use the graphing utility to draw the scatter diagram of the data and comment on the type of relationship that may exist between the two variables.

Plot the points in a cartesian coordinate system.

The scatter diagram is,

As we see the relationship between the two variables is represented by a parabola facing downwards.

The given table is,

Using the curve fitting through a graphing utility the equation that best fit the relation of the two variables are, $f\left(x\right)=-0.00371212x^2+1.03182x+5.66667.$

We need to use the function found in part (b) to determine how far the ball will travel before it reaches its maximum.

Let us determine the $x$- coordinate of the vertex to identify the horizontal distance at which maximum height is obtained.

$f\left(x\right)=-0.00371212x^2+1.03182x+5.66667.$

$x=-\frac{b}{2a}.$

$x=-\frac{1.03182}{2\left(-0.00371212\right)}.$

$≈138.98ft.$

At approximately$138.98ft$the ball will achieve maximum height.

The given table is,

We need to use the function found in part (b) to determine the maximum height of the ball.

Let us evaluate $f\left(138.98\right)$to obtain the maximum height attained by the ball.

$f\left(138.08\right)=-0.00371212{\left(138.98\right)}^2+1.03182\left(138.98\right)+5.66667.$

$≈77.37f{t}^2.$ $≈77.37f{t}^2.$

The maximum height attained by the ball is approximately$77.37f{t}^2.$

The given table is,

We need to graph the quadratic function of best fit on the scatter diagram with a graphing utility.

The graph of the quadratic function is