Question

Use the table to answer the question.

x Substituted Evaluate True Statement?
12
24
36
48
When set off, a certain firework follows the path of the quadratic function h=−25/36 x2 + 16 2/3x , 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= −25/36 x2 + 16 2/3x .

Answer 1

so the answer is 24?

Answer 2

To determine which value of \( x \) in the table is a solution to the equation

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

we can simplify the problem by substituting each \( x \) value from the table into the quadratic equation and checking if the resulting expression equals zero.

Let's rewrite the equation using decimals for easier calculations:

The equation can also be written as

\[
0 = -0.6944x^2 + 16.67x
\]

Now, substitute each \( x \) value from the table back into the equation:

1. **For \( x = 12 \)**:
\[
0 = -\frac{25}{36}(12^2) + \frac{50}{3}(12)
\]
\[
= -\frac{25}{36}(144) + \frac{50}{3}(12) = -100 + 200 = 100 \quad (\text{not } 0)
\]

2. **For \( x = 24 \)**:
\[
0 = -\frac{25}{36}(24^2) + \frac{50}{3}(24)
\]
\[
= -\frac{25}{36}(576) + \frac{50}{3}(24) = -400 + 400 = 0 \quad (\text{True Statement})
\]

3. **For \( x = 36 \)**:
\[
0 = -\frac{25}{36}(36^2) + \frac{50}{3}(36)
\]
\[
= -\frac{25}{36}(1296) + \frac{50}{3}(36) = -900 + 600 = -300 \quad (\text{not } 0)
\]

4. **For \( x = 48 \)**:
\[
0 = -\frac{25}{36}(48^2) + \frac{50}{3}(48)
\]
\[
= -\frac{25}{36}(2304) + \frac{50}{3}(48) = -1600 + 800 = -800 \quad (\text{not } 0)
\]

Based on the calculations, the only value of \( x \) that solves the equation \( 0 = -\frac{25}{36}x^2 + \frac{50}{3}x \) is:

\[
\boxed{24}
\]

The corresponding true statement for the table can be completed as follows:
- When \( x = 24 \), Substitute = 24, Evaluate = 0, True Statement? = Yes.

Answer 3

Yes, that's correct. The answer is \( x = 24 \). This is the value that satisfies the equation \( 0 = -\frac{25}{36}x^2 + \frac{50}{3}x \), meaning that at this point, the firework reaches the ground.