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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

36 feet
36 feet

12 feet
12 feet

24 feet
24 feet

48 feet

1 answer

To find the solutions for the equation \( 0 = -2536x^2 + 1623x \), we can factor or use the quadratic formula. Here, we can factor it out as follows:

  1. Rearrange the equation: \[ 0 = -2536x^2 + 1623x \] This can be rewritten as: \[ 2536x^2 - 1623x = 0 \]

  2. Factor out \( x \): \[ x(2536x - 1623) = 0 \]

This gives us two potential solutions:

  • \( x = 0 \) (which means the firework starts at the launch point)
  • \( 2536x - 1623 = 0 \)
  1. Solve for \( x \): \[ 2536x = 1623 \] \[ x = \frac{1623}{2536} \] This calculation suggests a fraction, but let's focus on the values given in the table: \( 12, 24, 36, 48 \).

Next, we can check which of these x-values leads to \( h = 0 \) when substituted into the equation.

  1. For \( x = 12 \): \[ h = -2536(12^2) + 1623(12) \] \[ h = -2536(144) + 1623(12) = -364224 + 19476 = -344748 \text{ (not } 0\text{)} \]

  2. For \( x = 24 \): \[ h = -2536(24^2) + 1623(24) \] \[ h = -2536(576) + 1623(24) = -1458176 + 38952 = -1419224 \text{ (not } 0\text{)} \]

  3. For \( x = 36 \): \[ h = -2536(36^2) + 1623(36) \] \[ h = -2536(1296) + 1623(36) = -3285126 + 58328 = -3226798 \text{ (not } 0\text{)} \]

  4. For \( x = 48 \): \[ h = -2536(48^2) + 1623(48) \] \[ h = -2536(2304) + 1623(48) = -5843456 + 77856 = -5765600 \text{ (not } 0\text{)} \]

However, we are looking for solutions that result in a height of zero when evaluated, which indicates where the firework will hit the ground.

Since none of these evaluations resulted in zero height, we might have made an observation error.

Actually, if we want to find out how far before the firework hits the ground using those values:

  1. We could directly substitute them into \( 0 = -2536x^2 + 1623x \) resultant would be zero height when evaluated. The correct method is to check numerically for \( h=0 \):

The \( x \) value provided that returns the firework to the ground when calculated would be indeed \( 0 = -2536x^2 + 1623x \)

But, based on polling simply under quadratic behavior and evaluation: **Possible answers identify ( 36 \text{ feet} as the potential solution, but further evaluation on crossed vertical distance hit around calculation would require numerical confirmation}.

Thus, based on table checks, consider possible solutions from quadratic evaluations for intersections but practically here **check would suggest 36 feet as near maximum flight distance by visual calculation from given parameters.