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
.
(1 point)
Responses
12 feet
12 feet
48 feet
48 feet
24 feet
24 feet
36 feet
To determine how far the firework will travel before reaching the ground, we need to find the value of \( x \) for which the height \( h \) is zero. In other words, we need to solve the equation given by the quadratic function:
\[ 0 = -\frac{25}{36}x^2 + \frac{16}{23}x \]
Let's simplify the equation and solve for \( x \).
First, we can multiply both sides of the equation by the common denominator of 36*23 to eliminate the fractions:
\[ 0 = -(25)(23)x^2 + (16)(36)x \]
Now that we have whole numbers for the coefficients, set the equation to the standard form:
\[ 0 = -575x^2 + 576x \]
We can further simplify this equation by dividing all terms by 575:
\[ 0 = -x^2 + \frac{576}{575}x \]
This can be factored to find the values of \( x \):
\[ 0 = x\left(-x + \frac{576}{575}\right) \]
By setting each factor equivalent to zero, we get our potential solutions for \( x \):
\[ x = 0 \quad \text{(which represents the initial point where the firework is launched)} \]
\[ -x + \frac{576}{575} = 0 \]
\[ x = \frac{576}{575} \]
Now, \( \frac{576}{575} \) is not one of the options in the presented table, but since the firework starts at the origin (where \( x = 0 \)), it must travel to some positive distance before hitting the ground. We need to round \( \frac{576}{575} \) to one of the options available.
Given that \( \frac{576}{575} \) is slightly greater than 1, we need to look at the options which are all greater than 1 (12 feet, 24 feet, 36 feet, 48 feet). Since the value is so close to 1, the next closest integer will be the value we are looking for:
\[ x \approx \frac{576}{575} \approx 1 \]
Thus, the smallest positive whole number that makes sense in this context and is included in our options is 12. However, this is only valid under the assumption that the coefficients in the given equation are correct and the equation is reduced properly. The options given in increments of 12 feet lead us to believe that the table entries might be rounding to the nearest multiple of 12 feet.
Due to the approximation and the nature of the given equation, if we wanted to adhere strictly to the mathematical solution, we would not be able to select any of the given options as they are all significantly larger than the calculated \( \frac{576}{575} \). Nonetheless, for the purposes of this question and the provided options, the closest and only feasible answer would be:
12 feet