Q. 26 - Precalculus Enhanced with Graphing Utilities

Question

Height of a Ball A shot-putter throws a ball at an inclination of ${45}^{\circ }$to the horizontal. The following data represent the height of the ball $h$ at the instant that it has traveled $x$ feet horizontally.

(a) Use a graphing utility to draw a scatter diagram of the data. Comment on the type of relationship that may exist between the two variables. 

(b) Use a graphing utility to find the quadratic function of best fit that models the relation between distance and height.

 (c) Use the function found in part (b) to determine how far the ball will travel before it reaches its maximum height. 

(d) Use the function found in part (b) to find the maximum height of the ball.

 (e) With a graphing utility, graph the quadratic function of best fit on the scatter diagram 

Step-by-Step Solution

Answer

(a) The scatter diagram is,

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

(c) Approximately $138.98 ft$ the ball will achieve maximum height.

(d) The maximum achieved by the ball is approximately $77.37 ft.$

(e) The graph is,

Part (a) Step 1. Given information

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. 

Step 2. Graphing

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.

Part (b) Step 1. Given information

The given table is,

We need to use a graphing utility to determine the quadratic function of best fit.

Step 2. Simplify

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

Part (c) Step 1. Given information

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

Step 2. Simplify

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

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

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

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

   $\approx 138.98 ft.$

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

Part (d) Step 1. Given information

The given table is,

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

Step 2. Simplify

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

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

               $\approx 77.37 f{t}^2.$  $\approx 77.37 f{t}^2.$

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

Part (e) Step 1. Given information

The given table is, 

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

Step 2. Graphing

The graph of the quadratic function is