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= -25/36 x^2+16 2/3x
To solve the equation 0 = -(25/36)x^2 + (16 2/3)x, we need to set the quadratic equation equal to zero and solve for x. Before we do that, let's simplify the equation by converting the mixed fraction to an improper fraction.
16 2/3 is equal to (16 * 3 + 2)/3 = (48 + 2)/3 = 50/3. So the equation becomes:
0 = -(25/36)x^2 + (50/3)x
To solve for x, we'll factor the equation or use the quadratic formula.
Since the quadratic is already set equal to zero, we can try to factor. If factoring is difficult, we can use the quadratic formula, which is:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = -(25/36), b = 50/3, and c = 0. Let's plug these values into the formula:
x = (-(50/3) ± √((50/3)^2 - 4*(-25/36)*0)) / (2*(-25/36))
Since c = 0, the equation simplifies to:
x = (-(50/3) ± √((50/3)^2)) / (2*(-25/36))
Now, let's compute (50/3)^2 and simplify further:
x = (-(50/3) ± √(2500/9)) / (-50/36)
The square root of 2500/9 is 50/3, so:
x = (-(50/3) ± (50/3)) / (-50/36)
This gives us two solutions:
x = (-(50/3) + (50/3)) / (-50/36) -> x = 0 / (-50/36) -> x = 0
x = (-(50/3) - (50/3)) / (-50/36) -> x = (-100/3) / (-50/36)
Let's calculate the second solution:
x = (-100/3) / (-50/36) -> x = (-100/3) * (36/-50) -> x = (-100*36)/(3*(-50))
Simplifying this, we get:
x = (-4 * 36) / (-50/3)
x = (-4 * 36 * 3) / (-50)
x = (-144 * 3) / (-50)
x = 432 / 50
Now we simplify the fraction:
x = 8.64
So, the two solutions to the equation are x = 0 and x = 8.64. However, since x represents the distance, the only meaningful physical solution representing how far the firework will travel before reaching the ground would be x = 8.64 (assuming the unit of distance is consistent with the units used in the equation).