During a water balloon fight, Louisa uses a slingshot to fire a water balloon over the fortified wall

Question

of the opponent.When fired from the slingshot, the balloon will follow the path
$h(x)=-x^{2}+12x$ 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 $0=-x^{2}+12x$ How far
will the balloon travel before it reaches the ground?

Answer 1

To determine how far the balloon will travel before it reaches the ground, we need to solve the equation \( h(x) = 0 \).

Starting with the given equation:

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

We can rearrange it to:

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

Next, we can factor out an \( x \):

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

Setting each factor to zero gives us:

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

The solutions indicate that the balloon reaches the ground at \( x = 0 \) feet (the starting point) and at \( x = 12 \) feet.

Therefore, the balloon will travel a total distance of

\[ \boxed{12} \text{ feet} \]

before it reaches the ground.