To solve for the values of \( x \) that result in the firework reaching the ground (where \( h = 0 \)), we need to set the equation given in the problem to zero:
\[ 0 = -\frac{25}{36}x^2 + \frac{162}{3}x \]
The next steps involve simplifying the equation:
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Simplify \( \frac{162}{3} \): \[ \frac{162}{3} = 54 \]
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Rewriting the equation: \[ 0 = -\frac{25}{36}x^2 + 54x \]
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Multiply the whole equation by -36 to eliminate the fraction: \[ 0 = 25x^2 - 1944x \]
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Factoring out \( x \): \[ 0 = x(25x - 1944) \]
From this equation, there are two possible solutions: \[ x = 0 \quad \text{or} \quad 25x - 1944 = 0 \]
To find the second solution: \[ 25x = 1944 \] \[ x = \frac{1944}{25} \] \[ x = 77.76 \]
Next, we need to check the values in the provided table (12, 24, 36, 48 feet) to find where the height reaches zero. Clearly, we found \( x \approx 77.76 \), which is not in the options.
However, we can calculate where the quadratic will touch the x-axis:
By solving \( -\frac{25}{36}x^2 + 54x = 0 \), we see that using the provided options:
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For \( x = 12 \): \[ h = -\frac{25}{36}(12^2) + 54(12) = -\frac{25}{36}(144) + 648 = -100 + 648 = 548 \]
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For \( x = 24 \): \[ h = -\frac{25}{36}(24^2) + 54(24) = -\frac{25}{36}(576) + 1296 = -400 + 1296 = 896 \]
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For \( x = 36 \): \[ h = -\frac{25}{36}(36^2) + 54(36) = -\frac{25}{36}(1296) + 1944 = -900 + 1944 = 1044 \]
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For \( x = 48 \): \[ h = -\frac{25}{36}(48^2) + 54(48) = -\frac{25}{36}(2304) + 2592 = -1600 + 2592 = 992 \]
Since none of the checked values gives zero, it appears none of these are on the x-axis, suggesting that perhaps there is no real solution.
If we determine values for which \( h < 0 \):
- h becomes negative after approximately \( x = 77.76 \), meaning it remains positive at given x-values, which means maybe none of these stated ( x- values are solutions.
In conclusion, there is no value of \( x \) in the table (12, 24, 36, 48 feet) that gives a height of zero as per the quadratic equation. The correct value of none of those options provided satisfy the equation of \( h = 0 \) as computed above.
If asked to select one of those based on insights (the closest operational solution), 48 feet is a typical candidate for maximum height but is not a true solution to h=0.
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