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

24 feet
24 feet

48 feet
48 feet

12 feet
12 feet

36 feet

1 answer

To find out how far the firework will travel before reaching the ground, we need to solve the equation:

\[ 0 = -2536x^2 + 1623x \]

This can be factored out as follows:

\[ 0 = x(-2536x + 1623) \]

From this equation, we have two potential solutions:

  1. \( x = 0 \)
  2. \( -2536x + 1623 = 0 \)

Now, let's solve for \( x \) in the second equation:

\[ -2536x + 1623 = 0 \]

\[ 2536x = 1623 \]

\[ x = \frac{1623}{2536} \approx 0.64 \]

Since this value does not match any of the options given (24 feet, 48 feet, 12 feet, 36 feet), we can check the quadratic equation again; however, the question wants us to check the provided options.

We'll substitute each of the options into the equation \( h = -2536x^2 + 1623x \) to see which results in \( h = 0 \):

  1. For \( x = 24 \):

\[ h = -2536(24^2) + 1623(24) \]

Calculating \( h \):

\[ h = -2536(576) + 1623(24) = -1462656 + 38952 = -1423704 \quad (\text{not 0}) \]

  1. For \( x = 48 \):

\[ h = -2536(48^2) + 1623(48) \]

Calculating \( h \):

\[ h = -2536(2304) + 1623(48) = -5843584 + 77808 = -5765776 \quad (\text{not 0}) \]

  1. For \( x = 12 \):

\[ h = -2536(12^2) + 1623(12) \]

Calculating \( h \):

\[ h = -2536(144) + 1623(12) = -364224 + 19476 = -344748 \quad (\text{not 0}) \]

  1. For \( x = 36 \):

\[ h = -2536(36^2) + 1623(36) \]

Calculating \( h \):

\[ h = -2536(1296) + 1623(36) = -3287296 + 58428 = -3228868 \quad (\text{not 0}) \]

None of the options you provided yield \( h = 0 \), which means we need to re-check with the correct values or consider further details about the options. However, if you want to use the quadratic formula method or check integer solutions between values.

Ultimately, since none of the options provided solve the equation \( h = 0 \) accurately, more context or corrected values may be required from your set of provided answers. If we need to just verify the provided potential solutions, then none meet the requirement to return to ground level at height \( h = 0 \).