To determine how far the firework will travel before reaching the ground, we need to find the values of \( x \) that satisfy the equation:
\[ 0 = -\frac{25}{36}x^2 + \frac{50}{3}x \]
We can factor this equation to find the solutions:
-
Set the equation to zero: \[ -\frac{25}{36}x^2 + \frac{50}{3}x = 0 \]
-
Factor out \( x \): \[ x \left(-\frac{25}{36}x + \frac{50}{3}\right) = 0 \] This gives us one solution as \( x = 0 \) (which represents the starting point).
-
Now solve for the other factor: \[ -\frac{25}{36}x + \frac{50}{3} = 0 \] Rearranging gives: \[ -\frac{25}{36}x = -\frac{50}{3} \] Multiply both sides by -1: \[ \frac{25}{36}x = \frac{50}{3} \]
-
Multiply both sides by \( \frac{36}{25} \): \[ x = \frac{50}{3} \cdot \frac{36}{25} \]
-
Calculate this value: \[ x = \frac{50 \times 36}{3 \times 25} = \frac{1800}{75} = 24 \]
Thus, we find that \( x = 24 \) feet is the solution to the equation \( 0 = -\frac{25}{36}x^2 + \frac{50}{3}x \).
So, the firework will reach the ground after traveling 24 feet.
Final Response: 24 feet