Question

A football player kicks a football downfield. The height of the football increases until it reaches a maximum height of 15 yards, 30 yards away from the player. A second kick is modeled by f(x)=−0.032x(x−50), where f is the height (in yards) and x is the horizontal distance (in yards). Compare the distances that the footballs travel before hitting the ground.

Answer 1

Answer:

kick 1 has travelled 15 + 15 = 30 yards before hitting the ground

so kick 2 travels 25 + 25 = 50 yards before hitting the ground

first kick reached 8 yards and 2nd kick reached 20 yards  

Explanation:

1st kick travelled 15 yards to reach maximum height of 8 yards

so, it has travelled 15 + 15 = 30 yards before hitting the ground

2nd kick is given by the equation

y (x) = -0.032x(x - 50)

$Y = 1.6 X - 0.032x^2$

we know that maximum height occurs is given as

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

$y =- \frac{1.6}{2(-0.032)} = 25$

and maximum height is

$y = 1.6\times 25 - 0.032\times 25^2$

y = 20

so kick 2 travels 25 + 25 = 50 yards before hitting the ground

first kick reached 8 yards and 2nd kick reached 20 yards

Answer 2

Answer:

The second ball traveled more distance, horizontally and vertically.

Explanation:

The given function is

$f(x)=-0.032x(x-50)$

To find the distances that the second football travels, we need to find the vertex of its movement, because it's movement has a parabola form. The quadratic expression is

$f(x)=-0.032x^{2} +1.6x$

Where $a=-0.032$ and $b=1.6$

The vertex has coordinates of $(h,k)$, where

$h=-\frac{b}{2a}$

Replacing values, we have

$h=-\frac{1.6}{2(-0.032)}=25$

Then, $k=f(h)$

$k=f(25)=-0.032(25)^{2} +1.6(25)\\k=-20+40=20$

Which means the maximum height of the second football is 20 yards. That means it travels 40 yards vertically.

Now, its horizontal distance can be found when $f(x)=0$

$0=-0.032x^{2} +1.6x\\0=x(-0.032x+1.6)\\x_{1}=0\\ -0.032x_{2} +16=0\\x_{2}=\frac{-16}{-0.032}\\ x_{2}=500$

So, its horizontal distance is 500 yards.

Comparing the distances between the footballs.

Ball 1

Horizontal distance of 30 yards.

Vertical distance of 30 yards.

Ball 2

Horizontal distance of 500 yards.

Vertical distance of 40 yards.

If we find their difference, it would be

Horizontal: 500 - 30 = 470 yards.

Vertical: 40 - 30 = 10 yards.

Therefore, the second ball traveled more distance, horizontally and vertically.