To determine how far the firework travels before reaching the ground, we need to find the value of \( x \) when the height \( h \) is equal to zero. We can set the quadratic function equal to zero:
\[ 0 = -\frac{25}{36}x^2 + 16\left(\frac{2}{3}x\right) \]
First, simplify the second term:
\[ 16\left(\frac{2}{3}x\right) = \frac{32}{3}x \]
Now the equation becomes:
\[ 0 = -\frac{25}{36}x^2 + \frac{32}{3}x \]
Next, multiply the entire equation by 36 to eliminate the fraction:
\[ 0 = -25x^2 + 12 \cdot 32x \]
Now calculate \( 12 \cdot 32 \):
\[ 0 = -25x^2 + 384x \]
Rearranging gives us a standard quadratic form:
\[ 25x^2 - 384x = 0 \]
Next, factor out \( x \):
\[ x(25x - 384) = 0 \]
Setting each factor equal to zero gives us the solutions:
- \( x = 0 \)
- \( 25x - 384 = 0 \)
Solving the second equation for \( x \):
\[ 25x = 384 \] \[ x = \frac{384}{25} \] \[ x = 15.36 \]
Thus, the firework will travel 15.36 feet before reaching the ground. The solutions to the equation \( 0 = -\frac{25}{36}x^2 + 16\left(\frac{2}{3}x\right) \) are \( x = 0 \) and \( x = 15.36 \).
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