When set off a certain firework follows the path of the quadratic function h = -25/36x^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 = -25/36x^2 + 16 2/3x.

A: 24 feet

B: 48 feet

C: 12 feet

D: 36 feet

1 answer

To determine how far the firework will travel before reaching the ground, we need to solve the equation:

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

This can be rewritten as:

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

To clear the fraction, we can multiply the entire equation by 36:

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

Now, we can rewrite the equation:

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

Next, we can factor out \( 25x \):

\[ 25x(x - 24) = 0 \]

This gives us two solutions:

  1. \( 25x = 0 \) which results in \( x = 0 \)
  2. \( x - 24 = 0 \) which results in \( x = 24 \)

Since the problem is asking for the value of \( x \) when the firework reaches the ground, we find that:

The firework will travel 24 feet before reaching the ground.

Thus, the correct answer is A: 24 feet.