Question

As Isla prepares to set off fireworks, she wants to ensure the spectators are a safe distance away. When shot in the air, one firework will follow the path h(x) = -x^2 + 15x, where h(x) is the height of the firework, and x is the horizontal distance it travels in feet. To determine how far the firework will travel before it reaches the ground, Isla sets the height equal to zero, and solves the following quadratic equation: 0 = -x^2 + 15x. . How far will the firework travel before it reaches the ground?

Answer 1

To determine how far the firework will travel before it reaches the ground, we need to solve the equation \( h(x) = 0 \), where \( h(x) \) is the height of the firework given by the equation:

\[
h(x) = -x^2 + 15x
\]

Setting \( h(x) \) equal to zero, we have:

\[
0 = -x^2 + 15x
\]

We can rearrange this equation to make it easier to solve:

\[
x^2 - 15x = 0
\]

Next, we factor the equation:

\[
x(x - 15) = 0
\]

Setting each factor equal to zero gives us:

1. \( x = 0 \)
2. \( x - 15 = 0 \) which simplifies to \( x = 15 \)

The solutions to the equation are \( x = 0 \) and \( x = 15 \). Since \( x = 0 \) represents the starting point (when the firework is launched), the distance the firework travels before it reaches the ground is:

\[
\boxed{15} \text{ feet}
\]