Question
Substituted Evaluate True Statement?
12
24
36
48
BACK
When set off, a certain firework follows the path of the quadratic function
h = - 25
36 22 + 16⅔2, 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
+ 16⅔2
(1 point)
• 36 feet
• 12 feet
• 24 feet
• 48 feet
12
24
36
48
BACK
When set off, a certain firework follows the path of the quadratic function
h = - 25
36 22 + 16⅔2, 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
+ 16⅔2
(1 point)
• 36 feet
• 12 feet
• 24 feet
• 48 feet
Answers
Answered by
GPT-4o mini
To determine how far the firework will travel before reaching the ground, we need to find the value of \( x \) that makes the height \( h = 0 \). Given the function:
\[
h = -25x^2 + 16\frac{2}{3}x
\]
First, we set \( h \) to 0:
\[
0 = -25x^2 + 16\frac{2}{3}x
\]
This can be rewritten as:
\[
-25x^2 + 16.67x = 0
\]
We can factor out an \( x \):
\[
x(-25x + 16.67) = 0
\]
This gives us two solutions:
1. \( x = 0 \) (which represents the starting point)
2. Solve the second part:
\(-25x + 16.67 = 0\)
To isolate \( x \), we can rearrange it:
\[
25x = 16.67
\]
\[
x = \frac{16.67}{25} = 0.667
\]
However, 0.667 is not one of the given options.
We made a mistake factoring since height couldn't be zero with simply taking out one \( x \).
Let's solve the equation:
\[
25x^2 - 16.67x = 0
\]
Now divide through by \( x \) (assuming \( x \neq 0 \)):
\[
25x - 16.67 = 0
\]
So:
\[
25x = 16.67
\]
Solving for \( x \):
\[
x = \frac{16.67}{25} \approx 0.67
\]
None of the choices (12 feet, 24 feet, 36 feet, 48 feet) work here, indicating a mistake in assumption or data given in the question's context.
Instead, to test each of the options given, we can substitute them into the original \( h \) formula and check when \( h = 0 \). The reasonable way is by checking directly:
Calculate each given value into \( h = -25x^2 + 16.67x \) and check for zero.
Starting at 12 feet:
\[
h(12) = -25(12^2) + 16.67(12)
\]
\[
= -25(144) + 16.67(12)
= -3600 + 200.04 = -3399.96 \text{ (not zero)}
\]
Now checking 24 feet:
\[
h(24) = -25(24^2) + 16.67(24)
\]
\[
= -25(576) + 16.67(24)
= -14400 + 400.08 = -13999.92 \text{ (not zero)}
\]
Checking 36 feet:
\[
h(36) = -25(36^2) + 16.67(36)
\]
\[
= -25(1296) + 16.67(36)
= -32400 + 600.12 = -31899.88 \text{ (not zero)}
\]
Finally checking 48 feet:
\[
h(48) = -25(48^2) + 16.67(48)
\]
\[
= -25(2304) + 16.67(48)
= -57600 + 800.16 = -56800 \text{ (not zero)}
\]
Given all are non-zero and showing negative height, this suggests none of these values would imply a ground level height of zero which contradicts the problem.
Please verify your formula or options. The assumptions may suggest no values accurately help determine and indicating to return ground height logically.
\[
h = -25x^2 + 16\frac{2}{3}x
\]
First, we set \( h \) to 0:
\[
0 = -25x^2 + 16\frac{2}{3}x
\]
This can be rewritten as:
\[
-25x^2 + 16.67x = 0
\]
We can factor out an \( x \):
\[
x(-25x + 16.67) = 0
\]
This gives us two solutions:
1. \( x = 0 \) (which represents the starting point)
2. Solve the second part:
\(-25x + 16.67 = 0\)
To isolate \( x \), we can rearrange it:
\[
25x = 16.67
\]
\[
x = \frac{16.67}{25} = 0.667
\]
However, 0.667 is not one of the given options.
We made a mistake factoring since height couldn't be zero with simply taking out one \( x \).
Let's solve the equation:
\[
25x^2 - 16.67x = 0
\]
Now divide through by \( x \) (assuming \( x \neq 0 \)):
\[
25x - 16.67 = 0
\]
So:
\[
25x = 16.67
\]
Solving for \( x \):
\[
x = \frac{16.67}{25} \approx 0.67
\]
None of the choices (12 feet, 24 feet, 36 feet, 48 feet) work here, indicating a mistake in assumption or data given in the question's context.
Instead, to test each of the options given, we can substitute them into the original \( h \) formula and check when \( h = 0 \). The reasonable way is by checking directly:
Calculate each given value into \( h = -25x^2 + 16.67x \) and check for zero.
Starting at 12 feet:
\[
h(12) = -25(12^2) + 16.67(12)
\]
\[
= -25(144) + 16.67(12)
= -3600 + 200.04 = -3399.96 \text{ (not zero)}
\]
Now checking 24 feet:
\[
h(24) = -25(24^2) + 16.67(24)
\]
\[
= -25(576) + 16.67(24)
= -14400 + 400.08 = -13999.92 \text{ (not zero)}
\]
Checking 36 feet:
\[
h(36) = -25(36^2) + 16.67(36)
\]
\[
= -25(1296) + 16.67(36)
= -32400 + 600.12 = -31899.88 \text{ (not zero)}
\]
Finally checking 48 feet:
\[
h(48) = -25(48^2) + 16.67(48)
\]
\[
= -25(2304) + 16.67(48)
= -57600 + 800.16 = -56800 \text{ (not zero)}
\]
Given all are non-zero and showing negative height, this suggests none of these values would imply a ground level height of zero which contradicts the problem.
Please verify your formula or options. The assumptions may suggest no values accurately help determine and indicating to return ground height logically.