Questions: When set off, a certain firework follows the path of the quadratic function h=-25/96 x^2+16 2/3 x, where: - h= the height of the firework in feet. - x= the horizontal distance it travels in feet To determine how far the firework will travel before reaching the ground, determine which value of x in table is a solution to the equation 0=-5/3 x^2+16 2/3 x (1 point) 12 feet 48 feet 36 feet 24 feet

Solution Steps

To determine how far the firework will travel before reaching the ground, we need to find the value of \( x \) that satisfies the equation \( 0 = -\frac{5}{3} x^2 + 16 \frac{2}{3} x \). This involves substituting each given \( x \) value from the table into the equation and checking if the equation holds true (i.e., equals zero). The correct \( x \) value will make the equation true.

We start with the quadratic equation given by: \[ 0 = -\frac{5}{3} x^2 + 16 \frac{2}{3} x \] This can be rewritten as: \[ 0 = -\frac{5}{3} x^2 + \frac{50}{3} x \]

We will substitute each value of \( x \) from the table into the equation to check which one satisfies it.

1. For \( x = 12 \): \[ 0 = -\frac{5}{3} (12)^2 + \frac{50}{3} (12) = -\frac{5}{3} \cdot 144 + \frac{50}{3} \cdot 12 = -240 + 200 = -40 \quad (\text{not a solution}) \]

2. For \( x = 24 \): \[ 0 = -\frac{5}{3} (24)^2 + \frac{50}{3} (24) = -\frac{5}{3} \cdot 576 + \frac{50}{3} \cdot 24 = -960 + 400 = -560 \quad (\text{not a solution}) \]

3. For \( x = 36 \): \[ 0 = -\frac{5}{3} (36)^2 + \frac{50}{3} (36) = -\frac{5}{3} \cdot 1296 + \frac{50}{3} \cdot 36 = -2160 + 600 = -1560 \quad (\text{not a solution}) \]

4. For \( x = 48 \): \[ 0 = -\frac{5}{3} (48)^2 + \frac{50}{3} (48) = -\frac{5}{3} \cdot 2304 + \frac{50}{3} \cdot 48 = -3840 + 800 = -3040 \quad (\text{not a solution}) \]

None of the values \( 12, 24, 36, \) or \( 48 \) satisfy the equation \( 0 = -\frac{5}{3} x^2 + 16 \frac{2}{3} x \). Therefore, there is no solution among the provided options.

Final Answer