To determine which value of \( x \) in the table is a solution to the equation \( 0 = -2536x^2 + 1623x \), we set the equation equal to zero:
\[ -2536x^2 + 1623x = 0 \]
We can factor out \( x \):
\[ x(-2536x + 1623) = 0 \]
This gives us two potential solutions:
- \( x = 0 \)
- \( -2536x + 1623 = 0 \)
Solving the second equation for \( x \):
\[ -2536x = -1623 \] \[ x = \frac{1623}{2536} \]
Now we can calculate \( x \):
\[ x \approx 0.640 \]
However, since we're interested in the specific values in the table (12, 24, 36, and 48 feet), we'll need to check which values satisfy the original equation by substituting each \( x \) into \( h \):
Evaluating \( h \) for each \( x \):
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For \( x = 12 \): \[ h = -2536(12^2) + 1623(12) \] \[ h = -2536(144) + 1623(12) = -364224 + 19476 = -344748 , (\text{not } 0) \]
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For \( x = 24 \): \[ h = -2536(24^2) + 1623(24) \] \[ h = -2536(576) + 1623(24) = -1451520 + 38952 = -1412568 , (\text{not } 0) \]
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For \( x = 36 \): \[ h = -2536(36^2) + 1623(36) \] \[ h = -2536(1296) + 1623(36) = -3281792 + 58428 = -3223364 , (\text{not } 0) \]
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For \( x = 48 \): \[ h = -2536(48^2) + 1623(48) \] \[ h = -2536(2304) + 1623(48) = -5830784 + 77824 = -5752960 , (\text{not } 0) \]
Final Result:
None of the evaluated points give a height of 0. To find out where \( h \) reaches 0, you would typically use the quadratic formula to solve \( -2536x^2 + 1623x = 0 \).
So, the answer is that none of the values provided (12, 24, 36, or 48 feet) leads to a true statement of \( h = 0 \) directly from our checks. The closest derived values or further evaluation would require numerical methods or graphical analysis. However, with the context provided, none of the aforementioned options are correct in this case.
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