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 reaching the ground, we need to find the value of x that makes the equation 0 = -2536x^2 + 1623x true.
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a), where the quadratic is in the form ax^2 + bx + c = 0.
In our equation, a = -2536, b = 1623, and c = 0.
Plugging these values into the quadratic formula:
x = (-1623 ± √(1623^2 - 4(-2536)(0))) / (2*(-2536))
Simplifying further:
x = (-1623 ± √(2633329)) / (-5072)
The value inside the square root, √(2633329), is equal to 1623. Therefore:
x = (-1623 ± 1623) / (-5072)
Now we have two possible solutions for x:
1) x = (1623 + 1623) / (-5072) = 3246 / (-5072)
2) x = (1623 - 1623) / (-5072) = 0
It doesn't make sense for the firework to travel a negative distance, so we can disregard the first solution. Therefore, the value of x that represents the horizontal distance the firework will travel before reaching the ground is x = 0.
This means that the firework will reach the ground when x = 0, or in other words, it will explode at the point of origin.