To determine which value of \( x \) in the table is a solution to the equation \( 0 = -\frac{25}{36} x^2 + 16 \frac{2}{3} x \), we need to evaluate the function \( h \) for each of the provided \( x \) values. A solution is a value of \( x \) that results in \( h = 0 \).
The equation can be rewritten as:
\[ 0 = -\frac{25}{36} x^2 + \frac{50}{3} x \]
Now, we can check each \( x \) value from the table:
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For \( x = 12 \): \[ h = -\frac{25}{36}(12^2) + \frac{50}{3}(12) \] \[ h = -\frac{25}{36}(144) + \frac{50}{3}(12) \] \[ h = -100 + 200 = 100 \quad (\text{not a solution}) \]
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For \( x = 24 \): \[ h = -\frac{25}{36}(24^2) + \frac{50}{3}(24) \] \[ h = -\frac{25}{36}(576) + \frac{50}{3}(24) \] \[ h = -400 + 400 = 0 \quad (\text{solution found}) \]
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For \( x = 36 \): \[ h = -\frac{25}{36}(36^2) + \frac{50}{3}(36) \] \[ h = -\frac{25}{36}(1296) + \frac{50}{3}(36) \] \[ h = -900 + 600 = -300 \quad (\text{not a solution}) \]
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For \( x = 48 \): \[ h = -\frac{25}{36}(48^2) + \frac{50}{3}(48) \] \[ h = -\frac{25}{36}(2304) + \frac{50}{3}(48) \] \[ h = -1600 + 800 = -800 \quad (\text{not a solution}) \]
The only value of \( x \) that results in the height \( h = 0 \) is:
24 feet.