Question

A soccer player who is 27 feet from a goal attempted to kick the ball in into the goal. the flight of the ball is modeled by a parabola. the ball reached a maximum height of 10 feet when it was 15 feet from the soccer player. the goal has a height of 8 feet. will the soccer ball land in the goal?

Answer 1

The ball will land inside the goal.

Explanation:

Given : The flight of the ball is modeled by a parabola.

To find : Will the soccer ball land in the goal?

Solution :

Let the equation of the parabola be,

`y=a(x-h)^2+k`

Where, (h,k) are the vertex of the parabola,

According to question,

A soccer player who is 27 feet from a goal.

Let A be the point where player stand i.e, (27,0)

The height of the goal is 8 feet.

Let B be the point of height of goal i.e, (0,8)

The ball reached a maximum height of 10 feet when it was 15 feet from the soccer player.

The distance from goal to the distance 15 feet away from the player is

27-15=12 feet

Let C be the point with maximum height i.e, (12,10)

C is the vertex of the parabola and A is the point on the parabola.

Then, The equation of parabola is

`y=a(x-12)^2+10`

Point A satisfy the equation,

`0=a(27-12)^2+10`

`0=a(15)^2+10`

`a=-(10)/(225)`

Substitute back in equation,

`y=-(10)/(225)(x-12)^2+10`

Now, we find the y-intercept of the equation i.e, x=0,

`y=-(10)/(225)(0-12)^2+10`

`y=-(10)/(225)(144)+10`

`y=3.6`

If the y-intercept is greater than 8 then the ball will land outside the goal.

If the y-intercept is less than 8 then the ball will land inside the goal.

As 3.6<8

Therefore, The ball will land inside the goal.

Refer the attached figure below.