To determine how far the firework travels before reaching the ground, we need to find the values of \( x \) that satisfy the equation \( 0 = -2536x^2 + 1623x \). This can be rewritten in standard quadratic form as:
\[ -2536x^2 + 1623x = 0 \]
Factoring out \( x \) gives us:
\[ x(-2536x + 1623) = 0 \]
This implies two solutions:
- \( x = 0 \) (which corresponds to the start of the firework launch)
- \( -2536x + 1623 = 0 \)
Now, we can solve for \( x \):
\[ -2536x + 1623 = 0 \ 2536x = 1623 \ x = \frac{1623}{2536} \]
Calculating \( \frac{1623}{2536} \) gives approximately \( 0.6398 \) feet, which is not one of the provided responses.
Given the choices of 12 feet, 24 feet, 36 feet, and 48 feet, we will substitute these values back into the original equation to check which one makes \( h = 0 \):
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For \( x = 12 \): \[ h = -2536(12)^2 + 1623(12) \] \[ h = -2536(144) + 1623(12) \] \[ h = -364224 + 19476 = -344748 \quad (\text{not zero}) \]
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For \( x = 24 \): \[ h = -2536(24)^2 + 1623(24) \] \[ h = -2536(576) + 1623(24) \] \[ h = -1460160 + 38952 = -1421208 \quad (\text{not zero}) \]
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For \( x = 36 \): \[ h = -2536(36)^2 + 1623(36) \] \[ h = -2536(1296) + 1623(36) \] \[ h = -3288960 + 58428 = -3230532 \quad (\text{not zero}) \]
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For \( x = 48 \): \[ h = -2536(48)^2 + 1623(48) \] \[ h = -2536(2304) + 1623(48) \] \[ h = -5832960 + 77808 = -5755152 \quad (\text{not zero}) \]
None of the substituted values make \( h = 0 \), meaning that the fireworks will not reach the ground at any of these given horizontal distances.
However, if we were to state which \( x \) is equal to reaching the height of 0 based on the evaluations, then these values are not directly derived from the equation being solved.
Thus, all tested \( x \) values yield negative heights, indicating the firework stays in the air without returning to ground at precisely those set distances, since no height evaluations returned true statements regarding touching ground. It means there is no solution among the provided options or the distances require further polynomial inputs.