Chapter 7.7, Problem 15P

a.

To determine

To find the maximum height of the ball.

Answer to Problem 15P

The maximum height of the ball is 40 ft.

Explanation of Solution

Given information:

The distance between the ball and the goal-post is 25 yard (105 ft). The ball is being kicked at $x=0$ and the goal-post at $x=105$ , then the equation of the ball’s path is $y=\frac{4}{3}x-\frac{x^2}{90}$ , where $y$ is the ball’s height in feet.

The standard form of a quadratic equation: $y=a{x}^2+bx+c$ .

So, the formula of the axis of symmetry: $x=\frac{-b}{2a}$ .

Here, $y=\frac{4}{3}x-\frac{x^2}{90}$ .

Now, comparing it with the standard form of the quadratic equation:

  $a=-\frac{1}{90},b=\frac{4}{3}\text{ and }c=0$ .

By applying the formula of the axis of symmetry:

  $x=\frac{-b}{2a}=\frac{-\frac{4}{3}}{2\cdot (-\frac{1}{90})}=\frac{4}{3}\times 45=60$ .

So, the maximum horizontal distance covered by the ball is 60 ft.

  $\therefore y=\frac{4}{3}x-\frac{x^2}{90}=\frac{4}{3}\times 60-\frac{{\left(60\right)}^2}{90}=80-40=40$ .

Hence,

The maximum height of the ball is 40 ft.

b.

To determine

To check whether the ball crosses the goal-post crossbar or not.

Answer to Problem 15P

Yes, the ball clears the goal-post crossbar by a margin of 7.5 ft.

Explanation of Solution

Given information:

The distance between the ball and the goal post is 25 yard (105 ft). The ball is being kicked at $x=0$ and the goal post at $x=105$ , then the equation of the ball’s path is $y=\frac{4}{3}x-\frac{x^2}{90}$ , where $y$ is the ball’s height in feet and the height of the goal-post crossbar is 10 ft.

When $x=105$ ft,

  $y=\frac{4}{3}x-\frac{x^2}{90}=\frac{4}{3}\times 105-\frac{{\left(105\right)}^2}{90}=140-122.5=17.5$ .

So, the height of the ball when it crosses the crossbar is 17.5 ft.

  $\therefore$ The height of the ball when it crosses the crossbar is greater than the height of the crossbar.

  $\therefore$ Extra height gained by the ball when it crosses the crossbar $=17.5-10=7.5$ ft.

Hence,

Yes, the ball clears the goal-post crossbar by a margin of 7.5 ft.