Question

When set off, a certain firework follows the path of the quadratic function h=−2536x2+1623x
ℎ
=
−
25
36
𝑥
2
+
16
2
3
𝑥
, where:

h=
ℎ
=
the height of the firework in feet.
x=
𝑥
=
the horizontal distance it travels in feet.
To determine how far the firework will travel before reaching the ground, determine which value of x in table is a solution to the equation 0=−2536x2+1623x
0
=
−
25
36
𝑥
2
+
16
2
3
𝑥

Answer 1

To determine how far the firework will travel before reaching the ground, we need to find the values of \( x \) for which \( h = 0 \). That means we will solve the equation:

\[
0 = -\frac{2536}{36}x^2 + \frac{1623}{36}x
\]

This is a quadratic equation in standard form \( ax^2 + bx + c = 0 \).

First, simplify the equation:

\[
0 = -2536x^2 + 1623x
\]

To solve for \( x \), we can factor out \( x \):

\[
0 = x(-2536x + 1623)
\]

Setting each factor equal to zero gives us:

1. \( x = 0 \)
2. \(-2536x + 1623 = 0\)

From the second factor, we solve for \( x \):

\[
-2536x + 1623 = 0 \\
2536x = 1623 \\
x = \frac{1623}{2536}
\]

Now, calculate \( \frac{1623}{2536} \):

Using a calculator or performing the division gives:

\[
x \approx 0.640
\]

Thus the firework will travel approximately \( 0.640 \) feet before hitting the ground besides the starting point at \( x = 0 \).

**Conclusion**: The firework travels a horizontal distance of approximately \( 0.640 \) feet before reaching the ground.