To determine the reasonable range for the height function \( h(t) = 120t - 16t^2 \), we need to analyze the quadratic function.
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Identify the function: The function represents the height of the ball over time and is a downward-opening parabola because the coefficient of \( t^2 \) is negative.
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Find the vertex: The maximum height (the vertex) can be found using the formula: \[ t = -\frac{b}{2a} \] where \( a = -16 \) and \( b = 120 \). Thus, \[ t = -\frac{120}{2 \times -16} = \frac{120}{32} = 3.75 \text{ seconds} \]
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Calculate the maximum height: Now substitute \( t = 3.75 \) back into the height function to find the maximum height: \[ h(3.75) = 120(3.75) - 16(3.75^2) \] First, calculate \( 3.75^2 = 14.0625 \): \[ h(3.75) = 120(3.75) - 16(14.0625) \] \[ h(3.75) = 450 - 225 = 225 \text{ feet} \]
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Determine the reasonable range: The ball starts at height \( h = 0 \) when \( t = 0 \) and reaches a maximum height of \( 225 \) feet at \( t = 3.75 \) seconds, after which it will fall back to height \( h = 0 \) when it hits the ground (as \( t \) increases to about \( 7.5 \) seconds).
Thus, the reasonable range for the height \( h \) is from \( 0 \) feet to \( 225 \) feet.
The correct response is: B \({y: 0 \leq y \leq 225}\)
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