When set off, a certain firework follows the path of the quadratic function h = −25/36 x^2 + 16 2/3x, where: h= the height of the firework in feet. 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 = −25/36 x^2 + 16 2/3x.

Question

A) 48 feet
B) 12 feet
C) 24 feet
D) 36 feet

Answer 1

To determine which value of \( x \) corresponds to the firework reaching the ground, we need to solve the equation:

\[ 0 = -\frac{25}{36} x^2 + \frac{50}{3} x \]

First, we can simplify our equation by multiplying all terms by 36 to eliminate the fraction:

\[ 0 = -25x^2 + 600x \]

Now we can factor out \( x \):

\[ 0 = x(-25x + 600) \]

This gives us two solutions:

1. \( x = 0 \) (the point when the firework is launched)
2. \( -25x + 600 = 0 \)

Solving the second equation for \( x \):

\[ -25x + 600 = 0
25x = 600
x = \frac{600}{25} = 24 \]

Thus, the firework will reach the ground when \( x = 24 \) feet.

Now let's look at the options given:

A) 48 feet
B) 12 feet
C) 24 feet
D) 36 feet

The correct answer is:

C) 24 feet