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 solve for x in the equation given by the quadratic function when the height h is 0. The equation given is:
0 = -2536x^2 + 1623x
To find the x-values that make this equation true, we can factor out an x, since one of the solutions will be x = 0 (the firework starts at the ground level).
x(-2536x + 1623) = 0
Here, we can see that either x = 0 or -2536x + 1623 = 0 provide valid solutions.
Solving for the second part, -2536x + 1623 = 0, we get:
2536x = 1623
x = 1623 / 2536
Calculating the value of x yields:
x ≈ 0.64 (approximately, as the exact result doesn't match one of the provided answers)
Since x = 0 corresponds to the initial position and doesn't represent the distance the firework travels before hitting the ground, we are interested in the second root. This root does not exactly match any of the options provided (12 feet, 48 feet, 24 feet, 36 feet), so there might be an error in the transcription of the equation or the options. Assuming we made a mistake in our decimal approximation, let's solve it more carefully for an exact fraction.
x = 1623 / 2536
The fraction cannot be reduced further since 1623 and 2536 have no common divisors. However, the options provided suggest that we should have an integer solution. Therefore, we need to ensure that there isn't a mistake in copying the equation or the possible answers.
If we assume the coefficients of the quadratic equation were meant to be divisors of the possible integer answers, we could attempt to substitute each provided option into the equation to check for a solution.
For example, with x = 12, the original equation becomes:
0 = -2536(12)^2 + 1623(12)
If x = 12 is indeed the correct solution, this equation would yield 0. The equation will help us to determine if any of the other options (48, 24, 36 feet) could be the correct distance as well.
Let's test each option:
For x = 12:
0 = -2536(12)^2 + 1623(12)
0 = -2536(144) + 1623(12)
0 = -365184 + 19476
This does not yield 0, so x = 12 is not a solution.
For x = 48:
0 = -2536(48)^2 + 1623(48)
0 = -2536(2304) + 1623(48)
0 = -5845504 + 77808
This does not yield 0, so x = 48 is not a solution.
For x = 24:
0 = -2536(24)^2 + 1623(24)
0 = -2536(576) + 1623(24)
0 = -1460736 + 38904
This does not yield 0, so x = 24 is not a solution.
For x = 36:
0 = -2536(36)^2 + 1623(36)
0 = -2536(1296) + 1623(36)
0 = -3285504 + 58368
This does not yield 0, so x = 36 is not a solution.
None of the provided options are solutions to the equation 0 = -2536x^2 + 1623x. Thus, we either made a calculation error or there may be some mistake in the transcription of the problem or the options provided.