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

Question

ℎ
(
𝑥
)
=
−
𝑥
2
+
12
𝑥
, whereh(x)
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
0
=
−
𝑥
2
+
12
𝑥
. 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 set the height function \( h(x) \) equal to zero and solve for \( x \).

We start with the equation:

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

This can be rewritten as:

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

Factoring out \( x \):

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

This gives us two solutions:

1. \( x = 0 \)
2. \( -x + 12 = 0 \) (which can be rewritten as \( x = 12 \))

The \( x = 0 \) solution represents the point where the balloon is launched, and the \( x = 12 \) solution represents the point where the balloon hits the ground.

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