To determine how far the firework will travel before reaching the ground, we need to find the value of \( x \) where the height \( h \) is \( 0 \). This requires us to solve the equation:
\[ 0 = -\frac{25}{36}x^2 + 16\left(\frac{2}{3}x\right) \]
First, simplify the right side:
\[ 16\left(\frac{2}{3}x\right) = \frac{32}{3}x \]
So, the equation becomes:
\[ 0 = -\frac{25}{36}x^2 + \frac{32}{3}x \]
To eliminate the fractions, multiply the entire equation by \( 36 \), the least common multiple of the denominators:
\[ 0 = -25x^2 + 384x \]
This simplifies to:
\[ 25x^2 - 384x = 0 \]
Now, factor out \( x \):
\[ x(25x - 384) = 0 \]
This gives us two solutions:
- \( x = 0 \) (the point of launch)
- \( 25x - 384 = 0 \)
We will solve for \( x \) in the second equation:
\[ 25x = 384 \] \[ x = \frac{384}{25} \] \[ x = 15.36 \]
It seems there was an oversight in my computation compared to the options given. Let’s check the value:
\[ \frac{384}{25} \approx 15.36 \]
This value doesn’t match any of the provided options (A. 38, B. 24, C. 12, D. 48).
Let's try computing the distance in a different method for clarity:
- \( x(25x - 384) = 0 \)
- So, \( 25x = 384 \) yields \( x = 15.36 \).
Now let’s check the options against \( 25x - 384 = 0 \):
To check for further intersections with \( 0 \), solve the quadratic using the quadratic formula:
- The equation in standard form is: \[ -\frac{25}{36}x^2 + \frac{32}{3}x = 0 \]
This effectively means you can adjust the focus against the root option matrix for these solutions.
Clearly observable: \[ 25 = \frac{384}{x} \Rightarrow \text{ Indicating horizontal adjustment. } \]
Given the available options seem off, the primary outcome matches \( 15.36 \).
Thus, it appears there's a simplification error. But the proper maximum travel calculated did not yield a precise match within provided options. Validate further iterations on presented options as a comparison to peak. If consistent travel outputs land outside, ensuring final intersects resolve on trajectory choices within the model should be engaged.
Conclusively, further recheck or inquire sections for correct inputs as they may vary direct inputs from analytical sources or prior approximations.
However, from a quadratic solution perspective applied above: Final answer cannot align on options provided without further exploration.