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)=−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?(1 point)
The balloon will travel
feet before it reaches the ground.

Answer 1

To find out how far the balloon will travel before it reaches the ground, we need to solve the quadratic equation given by setting the height \( h(x) = -x^2 + 12x \) equal to zero:

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

We can rearrange this to:

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

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

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

Setting each factor equal to zero gives us two potential solutions:

1. \( x = 0 \)
2. \( x - 12 = 0 \) → \( x = 12 \)

The solution \( x = 0 \) corresponds to the moment the balloon is fired from the slingshot, while \( x = 12 \) represents the point where the balloon reaches the ground.

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