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 water balloon will travel before it reaches the ground, we need to solve the quadratic equation given by the height function \( h(x) = -x^2 + 12x \) when \( h(x) \) is set to zero.

Setting the equation to zero:

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

To make it easier to solve, we can factor out \( x \):

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

This gives us two factors:

1. \( x = 0 \) (which represents the starting point of the balloon being fired)
2. \( -x + 12 = 0 \) → \( x = 12 \)

The solutions to the equation are \( x = 0 \) and \( x = 12 \). The value \( x = 12 \) represents the horizontal distance the balloon travels before it hits the ground.

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