Question
When set off, a certain firework follows the path of the quadratic function h= -25/36x^2 + 16(2/3x)
X= the horizontal distance it travels
H= the height of the firework in feet
Determine how far the firework will travel before reaching the ground, determine the value that is the solution to the equation 0= -25/36x^2 + 16(2/3x)
A. 38
B. 24
C. 12
D. 48
X= the horizontal distance it travels
H= the height of the firework in feet
Determine how far the firework will travel before reaching the ground, determine the value that is the solution to the equation 0= -25/36x^2 + 16(2/3x)
A. 38
B. 24
C. 12
D. 48
Answers
GPT-4o mini
To determine how far the firework will travel before reaching the ground, we need to find the value of \( x \) where the height \( h \) is \( 0 \). This requires us to solve the equation:
\[
0 = -\frac{25}{36}x^2 + 16\left(\frac{2}{3}x\right)
\]
First, simplify the right side:
\[
16\left(\frac{2}{3}x\right) = \frac{32}{3}x
\]
So, the equation becomes:
\[
0 = -\frac{25}{36}x^2 + \frac{32}{3}x
\]
To eliminate the fractions, multiply the entire equation by \( 36 \), the least common multiple of the denominators:
\[
0 = -25x^2 + 384x
\]
This simplifies to:
\[
25x^2 - 384x = 0
\]
Now, factor out \( x \):
\[
x(25x - 384) = 0
\]
This gives us two solutions:
1. \( x = 0 \) (the point of launch)
2. \( 25x - 384 = 0 \)
We will solve for \( x \) in the second equation:
\[
25x = 384
\]
\[
x = \frac{384}{25}
\]
\[
x = 15.36
\]
It seems there was an oversight in my computation compared to the options given. Let’s check the value:
\[
\frac{384}{25} \approx 15.36
\]
This value doesn’t match any of the provided options (A. 38, B. 24, C. 12, D. 48).
Let's try computing the distance in a different method for clarity:
1. \( x(25x - 384) = 0 \)
2. So, \( 25x = 384 \) yields \( x = 15.36 \).
Now let’s check the options against \( 25x - 384 = 0 \):
To check for further intersections with \( 0 \), solve the quadratic using the quadratic formula:
- The equation in standard form is:
\[
-\frac{25}{36}x^2 + \frac{32}{3}x = 0
\]
This effectively means you can adjust the focus against the root option matrix for these solutions.
Clearly observable:
\[
25 = \frac{384}{x} \Rightarrow \text{ Indicating horizontal adjustment. }
\]
Given the available options seem off, the primary outcome matches \( 15.36 \).
**Thus, it appears there's a simplification error. But the proper maximum travel calculated did not yield a precise match within provided options. Validate further iterations on presented options as a comparison to peak. If consistent travel outputs land outside, ensuring final intersects resolve on trajectory choices within the model should be engaged.**
Conclusively, further recheck or inquire sections for correct inputs as they may vary direct inputs from analytical sources or prior approximations.
However, from a quadratic solution perspective applied above:
**Final answer cannot align on options provided without further exploration.**
\[
0 = -\frac{25}{36}x^2 + 16\left(\frac{2}{3}x\right)
\]
First, simplify the right side:
\[
16\left(\frac{2}{3}x\right) = \frac{32}{3}x
\]
So, the equation becomes:
\[
0 = -\frac{25}{36}x^2 + \frac{32}{3}x
\]
To eliminate the fractions, multiply the entire equation by \( 36 \), the least common multiple of the denominators:
\[
0 = -25x^2 + 384x
\]
This simplifies to:
\[
25x^2 - 384x = 0
\]
Now, factor out \( x \):
\[
x(25x - 384) = 0
\]
This gives us two solutions:
1. \( x = 0 \) (the point of launch)
2. \( 25x - 384 = 0 \)
We will solve for \( x \) in the second equation:
\[
25x = 384
\]
\[
x = \frac{384}{25}
\]
\[
x = 15.36
\]
It seems there was an oversight in my computation compared to the options given. Let’s check the value:
\[
\frac{384}{25} \approx 15.36
\]
This value doesn’t match any of the provided options (A. 38, B. 24, C. 12, D. 48).
Let's try computing the distance in a different method for clarity:
1. \( x(25x - 384) = 0 \)
2. So, \( 25x = 384 \) yields \( x = 15.36 \).
Now let’s check the options against \( 25x - 384 = 0 \):
To check for further intersections with \( 0 \), solve the quadratic using the quadratic formula:
- The equation in standard form is:
\[
-\frac{25}{36}x^2 + \frac{32}{3}x = 0
\]
This effectively means you can adjust the focus against the root option matrix for these solutions.
Clearly observable:
\[
25 = \frac{384}{x} \Rightarrow \text{ Indicating horizontal adjustment. }
\]
Given the available options seem off, the primary outcome matches \( 15.36 \).
**Thus, it appears there's a simplification error. But the proper maximum travel calculated did not yield a precise match within provided options. Validate further iterations on presented options as a comparison to peak. If consistent travel outputs land outside, ensuring final intersects resolve on trajectory choices within the model should be engaged.**
Conclusively, further recheck or inquire sections for correct inputs as they may vary direct inputs from analytical sources or prior approximations.
However, from a quadratic solution perspective applied above:
**Final answer cannot align on options provided without further exploration.**