Question
Once that off a certain part work follows the path of the quadratic function h equals 25 over 36x squared plus 16 2/3x or h equals the height of the firework in feet x equals 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 the table is a solution to the equation zero results -25/36 x^2 + 16 2/3 x
Answers
GPT-4o mini
To determine the horizontal distance at which the firework will reach the ground, we need to find the value of \( x \) for which the height \( h \) equals 0. The equation provided is:
\[
h = \frac{25}{36}x^2 + \frac{50}{3}x
\]
First, let’s set \( h = 0 \):
\[
0 = \frac{25}{36}x^2 + \frac{50}{3}x
\]
Now, to eliminate the fractions, we can multiply through by 36 (the least common multiple of the denominators):
\[
0 = 25x^2 + 600x
\]
Now we have a quadratic equation:
\[
25x^2 + 600x = 0
\]
Factoring out the common term:
\[
25x(x + 24) = 0
\]
Setting each factor to zero gives:
1. \( 25x = 0 \) ⇒ \( x = 0 \)
2. \( x + 24 = 0 \) ⇒ \( x = -24 \)
Since we are looking for a horizontal distance, the only valid \( x \) value is \( x = 0\).
If the table you mention includes positive values for \( x \), you would only consider the initial point of launch, \( x = 0 \). To find how far it travels before reaching the ground (positive values), you would consider next calculating how far it reaches at the apex or its maximum height if you have more data or equations to follow. If you have a specific table or additional values provided in a table format, please provide them, and I can assist further.
\[
h = \frac{25}{36}x^2 + \frac{50}{3}x
\]
First, let’s set \( h = 0 \):
\[
0 = \frac{25}{36}x^2 + \frac{50}{3}x
\]
Now, to eliminate the fractions, we can multiply through by 36 (the least common multiple of the denominators):
\[
0 = 25x^2 + 600x
\]
Now we have a quadratic equation:
\[
25x^2 + 600x = 0
\]
Factoring out the common term:
\[
25x(x + 24) = 0
\]
Setting each factor to zero gives:
1. \( 25x = 0 \) ⇒ \( x = 0 \)
2. \( x + 24 = 0 \) ⇒ \( x = -24 \)
Since we are looking for a horizontal distance, the only valid \( x \) value is \( x = 0\).
If the table you mention includes positive values for \( x \), you would only consider the initial point of launch, \( x = 0 \). To find how far it travels before reaching the ground (positive values), you would consider next calculating how far it reaches at the apex or its maximum height if you have more data or equations to follow. If you have a specific table or additional values provided in a table format, please provide them, and I can assist further.