Chapter 3.4, Problem 26AYU

Height of a Rail A shot-putter throws a hall at an inclination of ${45}^{\circ }$ to the horizontal. The following data represent the height of the ball $h$, in feet, 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 relation 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.

To determine

To calculate:

a. Graph a scattered diagram and determine the type of relation that exists between the 2 variables.

b. Find the quadratic function of best fit.

c. Determine how far the ball will travel before it reaches the maximum height.

d. Find the maximum height of the ball.

e. Graph the quadratic function of best fit.

Answer to Problem 26AYU

a. The graph is given below.

b. The equation of best fit is $y=-0.0037x^2+1.0318x+5.6667$.

c. The ball will travel $139.43$ feet before it reaches the maximum height.

d. The maximum height attained by the ball is about $66.27\text{ feet}$.

e. The graph is given below.

Explanation of Solution

Given:

A shot putter throws a ball at an inclination of 45 degree to the horizontal. The given data represents the relation between the height of the ball thrown and the distance it travels.

Formula used:

For a quadratic equation $y=a{x}^2+bx+c,<mtext> </mtext>$ if $a$ is positive, then the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the minimum point and if $a$ is negative, the vertex $\left(\frac{-b}{2a},<mtext> </mtext>y\left(\frac{-b}{2a}\right)\right)$ is the maximum point.

Calculation:

We can draw the scatter diagram using Microsoft Excel.

Thus, on entering the $x$ and the $y$ values on excel, we have to choose the Scatter diagram form the insert option.

Then for getting the equation of best fit, we have to choose the Layout option and then Trendline and then more Trendline option. Then choose the option polynomial and then display the equation.

Thus, we get the scatter diagram with the equation of best fit as

a. The scatter diagram is drawn above and we can see that the given variables exhibit a non-linear polynomial relationship.

b. The equation of best fit is $y=-0.0037x^2+1.0318x+5.6667$.

c. We can see that the equation of best fit is a quadratic equation with $a=-0.0037,<mtext> </mtext>b=1.0318,<mtext> </mtext>c=5.6667$.

Here, $a$ is negative, therefore, the vertex is the maximum point.

The $x$ value for maximum $y$ is

$\begin{array}{l}x=-\frac{b}{2a}=-\frac{1.0318}{2(-0.0037)}\end{array}$

$=139.43$

Thus, the ball will travel $139.43$ feet before it reaches the maximum height.

d. The maximum income earned is

$\begin{array}{l}y(139.43)=-0.0037(139.43)<msup>\; <mi></mi>\; <mn>2</mn>\; </msup>+1.0318(139.43)+5.6667\end{array}$

$=66.27$

The maximum height attained by the ball is about $66.27$ feet.

e. The graph is drawn above.