A carnival game requires that two players stand five feet apart and throw a ball toward each other. If they are able to make the balls collide, they win. The height reached by Elio’s ball follows a trajectory modeled by y=−x2+3x+6

Question

, and the height reached by Jhumpa’s ball follows a trajectory modeled by y=−(x−3)2+9
. At what height do the balls collide?(1 point)
Responses

8.25 feet
8.25 feet

2 feet
2 feet

8 feet
8 feet

9 feet

Answer 1

To find the height at which Elio's and Jhumpa's balls collide, we need to set their height equations equal to each other and solve for \( x \).

Elio's ball trajectory:

\[ y = -x^2 + 3x + 6 \]

Jhumpa's ball trajectory:

\[ y = -(x - 3)^2 + 9 \]

First, we can expand Jhumpa's equation:

\[ y = -(x^2 - 6x + 9) + 9 \] \[ y = -x^2 + 6x - 9 + 9 \] \[ y = -x^2 + 6x \]

Now, we set the two equations equal to each other:

\[ -x^2 + 3x + 6 = -x^2 + 6x \]

Next, we can eliminate \(-x^2\) from both sides:

\[ 3x + 6 = 6x \]

Subtract \(3x\) from both sides:

\[ 6 = 3x \]

Divide both sides by \(3\):

\[ x = 2 \]

Now, we can substitute \(x = 2\) back into either equation to find the height \(y\). We can use Elio's equation:

\[ y = -2^2 + 3(2) + 6 \] \[ y = -4 + 6 + 6 \] \[ y = 8 \]

Thus, the balls collide at a height of \( \boxed{8} \) feet.