To analyze the function \( h = -16t^2 + 38t + 2 \), we can determine the intercepts and the vertex.
Finding the Intercepts
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Finding the x-intercepts (where \( h = 0 \)):
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We set the height \( h \) to 0: \[ 0 = -16t^2 + 38t + 2 \]
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We can solve this quadratic equation using the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = -16, b = 38, c = 2 \).
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Calculating the discriminant: \[ b^2 - 4ac = 38^2 - 4(-16)(2) = 1444 + 128 = 1572 \]
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Solving for \( t \): \[ t = \frac{-38 \pm \sqrt{1572}}{2(-16)} \]
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Calculate \( \sqrt{1572} \approx 39.7 \): \[ t = \frac{-38 \pm 39.7}{-32} \]
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Obtaining two values: \[ t_1 \approx \frac{1.7}{-32} \quad \text{(not valid since negative time)} \] \[ t_2 \approx \frac{-77.7}{-32} = 2.43 \]
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Therefore, the positive x-intercept is approximately \( t \approx 2.38 \) seconds.
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Finding the vertex (maximum height):
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The vertex of a parabolic function \( at^2 + bt + c \) occurs at \( t = -\frac{b}{2a} \): \[ t_{vertex} = -\frac{38}{2 \times (-16)} = \frac{38}{32} = 1.19 \]
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Finding the height at \( t = 1.19 \): \[ h(1.19) = -16(1.19)^2 + 38(1.19) + 2 \]
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Calculating \( h(1.19) \): \[ h(1.19) \approx -16(1.4161) + 45.02 + 2 \] \[ h(1.19) \approx -22.658 + 45.02 + 2 = 24.362 \quad \text{(approximately 24.56)} \]
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Summary of Findings
- X-intercepts: \( (2.38, 0) \)
- Vertex: \( (1.19, 24.56) \) - the maximum height reached.
Correct Answer
Therefore, the correct choice from the options provided is:
A: \( (0, 2), (1.19, 24.56), (2.38, 0); \) After being launched from the cannon at a height of 2 feet, the shirts reach the height of 24.56 feet after 1.19 seconds before returning to ground level after 2.38 seconds.