Question

During a water balloon fight, Louisa uses a slingshot to fire a water balloon over the fortified wall of the opponent. When fired from the slingshot, the balloon will follow the path h(x)=−x2+12x
, where h(x)
is the height of the balloon in feet, and x is the horizontal distance it will travel. To determine how far the balloon will travel before it reaches the ground, Louisa sets the height equal to zero and solves the quadratic equation 0=−x2+12x
. How far will the balloon travel before it reaches the ground?

Answer 1

To determine how far the water balloon will travel before it reaches the ground, we need to solve the quadratic equation given by the height function set equal to zero:

\[
0 = -x^2 + 12x
\]

First, we rearrange the equation:

\[
-x^2 + 12x = 0
\]

To solve this, we can factor the equation. We can factor out \( -x \):

\[
-x(x - 12) = 0
\]

This gives us two potential solutions:

1. \( -x = 0 \), which simplifies to \( x = 0 \)
2. \( x - 12 = 0 \), which simplifies to \( x = 12 \)

The first solution \( x = 0 \) represents the initial launch position of the water balloon, while the second solution \( x = 12 \) represents the distance traveled before the balloon reaches the ground.

Therefore, the balloon will travel **12 feet** before it reaches the ground.