Use the table to answer the question.

Question

x Substituted Evaluate True Statement?
12
24
36
48
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 2/3 x
.

(1 point)
Responses

48 feet
48 feet

36 feet
36 feet

24 feet
24 feet

12 feet
12 feet

Answer 1

To determine how far the firework will travel before reaching the ground, we need to solve the equation \(0 = -\frac{25}{36}x^2 + \frac{50}{3}x\) (as \(16 \frac{2}{3}\) can be converted to \(\frac{50}{3}\)) for the values of \(x\) in the table.

We can set up the equation:

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

Factoring out \(x\):

\[ x \left(-\frac{25}{36}x + \frac{50}{3}\right) = 0 \]

This gives us two possible solutions:

1. \(x = 0\) (which represents when the firework is launched)
2. Solving for the second factor:

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

Multiplying through by \(-36\) to eliminate the fraction:

\[ 25x = -600 \]

Now, isolate \(x\):

\[ 25x = 600 \quad \Rightarrow \quad x = \frac{600}{25} = 24 \]

So the height of the firework reaches the ground again when \(x = 24\) feet.

Thus, the correct response from the options is:

24 feet.