Question

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
, 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.