When set off, a certain firework follows the path of the quadratic function h=-25/36x62 + 16 2/3x, where :h = the height of the firework in feet and x=the horizontal distance in travels in feet

Question

to determine how far the firework will travel before reaching the ground, determine which value of x in the table is a solution to the equation 0=-25/36x62 + 16 2/3x
Values of x in the table are 12, 24, 36, 48

Answer 1

To determine which value of $x$ corresponds to the firework reaching the ground, we need to solve the equation:

$$0=-\frac{25}{36}{x}^2+\frac{50}{3}x$$

This simplifies to solving:

$$\frac{25}{36}{x}^2-\frac{50}{3}x=0$$

Factoring out $x$:

$$x(-\frac{25}{36}x+\frac{50}{3})=0$$

Setting each factor to zero, we have:

1. $x=0$
2. $-\frac{25}{36}x+\frac{50}{3}=0$

Let's solve the second equation:

$$-\frac{25}{36}x+\frac{50}{3}=0$$

Rearranging gives:

$$\frac{25}{36}x=\frac{50}{3}$$

Multiplying both sides by $\frac{36}{25}$:

$$x=\frac{50}{3}\cdot \frac{36}{25}$$

Calculating the right-hand side:

$$x=\frac{50\times 36}{3\times 25}=\frac{1800}{75}=24$$

Thus, $x=24$ is the value where the firework will reach the ground.

Checking the table’s values (12, 24, 36, 48), the solution to the equation is:

$$\boxed{{\displaystyle 24}}$$