To determine the horizontal distance at which the firework will reach the ground, we need to find the value of \( x \) for which the height \( h \) equals 0. The equation provided is:
\[ h = \frac{25}{36}x^2 + \frac{50}{3}x \]
First, let’s set \( h = 0 \):
\[ 0 = \frac{25}{36}x^2 + \frac{50}{3}x \]
Now, to eliminate the fractions, we can multiply through by 36 (the least common multiple of the denominators):
\[ 0 = 25x^2 + 600x \]
Now we have a quadratic equation:
\[ 25x^2 + 600x = 0 \]
Factoring out the common term:
\[ 25x(x + 24) = 0 \]
Setting each factor to zero gives:
- \( 25x = 0 \) ⇒ \( x = 0 \)
- \( x + 24 = 0 \) ⇒ \( x = -24 \)
Since we are looking for a horizontal distance, the only valid \( x \) value is \( x = 0\).
If the table you mention includes positive values for \( x \), you would only consider the initial point of launch, \( x = 0 \). To find how far it travels before reaching the ground (positive values), you would consider next calculating how far it reaches at the apex or its maximum height if you have more data or equations to follow. If you have a specific table or additional values provided in a table format, please provide them, and I can assist further.
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