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?

A: -15 feet

B: 15 feet

C: 30 feet

D: 7.5 feet

Answer 1

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

The height of the firework is given by:

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

Setting the height equal to zero:

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

We can rearrange this to:

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

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

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

Setting each factor to zero gives us the solutions:

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

The solutions \( x = 0 \) and \( x = 15 \) represent the points in time when the firework is at ground level (height = 0). The firework is launched at \( x = 0 \) and returns to the ground at \( x = 15 \).

Thus, the distance the firework travels before it reaches the ground is:

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

So the correct answer is **B: 15 feet**.