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=−2536x2+1623x
, 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=−2536x2+1623x
.
To determine how far the firework will travel before it reaches the ground, we want to find the value of \( x \) where the height (\( h \)) is equal to 0. The corresponding equation is:
\[ 0 = -\frac{25}{36}x^2 + \frac{16}{23}x \]
Let's solve this quadratic equation for \( x \).
To make things easier, we can multiply both sides of the equation by 36 * 23 to get rid of the fractions:
\[ 0 = -25 \cdot 23 \cdot x^2 + 16 \cdot 36 \cdot x \]
\[ 0 = -575x^2 + 576x \]
Now, we can factor out an \( x \), as the equation appears to be factorable:
\[ 0 = x(-575x + 576) \]
This gives us two possible solutions: \( x = 0 \) (when the firework is launched) and \( x = \frac{576}{575} \), which is when the firework hits the ground after being launched (ignoring the initial \( x = 0 \)).
So, \( x = \frac{576}{575} \) feet is the non-zero horizontal distance at which the firework will land. However, we want to compare this to the whole numbers given in the table (12, 24, 36, 48).
Upon closer inspection, there seems to be an error in the simplification. Let's correct it:
The correct simplification from the original equation should be:
\[ 0 = x(-\frac{25}{36}x + \frac{16}{23}) \]
Since we're looking for the positive solution where the firework hits the ground, we can set the second factor equal to zero and solve for \( x \):
\[ -\frac{25}{36}x + \frac{16}{23} = 0 \]
First, multiply both sides by \( 36 \cdot 23 \) to eliminate fractions:
\[ 0 = -25 \cdot 23 \cdot x + 16 \cdot 36 \]
Factor out the coefficients to find the common multiple:
\[ 0 = -(25 \cdot 23) \cdot x + (16 \cdot 36) \]
Now, add \( 25 \cdot 23 \cdot x \) to both sides:
\[ (25 \cdot 23) \cdot x = 16 \cdot 36 \]
Divide both sides by \( 25 \cdot 23 \):
\[ x = \frac{16 \cdot 36}{25 \cdot 23} \]
Now, let's calculate the exact value of \( x \):
\[ x = \frac{16 \cdot 36}{25 \cdot 23} = \frac{576}{575} \]
Notice that \( \frac{576}{575} \) is just a little over 1, which means it doesn't match any of the whole number values in the table (12, 24, 36, 48).
Since the question specifies we have to choose from the given values of \( x \) in the table, none of the values in the table are the exact solutions to the equation. The firework will reach the ground at a horizontal distance slightly over 1 foot, not at 12, 24, 36, or 48 feet. Therefore, none of the values of \( x \) in the table are true solutions for the given equation, and we can't determine the answer to the question using the provided table values.