To analyze the problem and find the intercepts, vertex, and path of the shirts, we'll start with the provided equation for height \( h(t) = -16t^2 + 38t + 2 \), where the initial height of the cannon is 2 feet.
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Finding the intercepts:
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The vertical intercept ( h-intercept ) is when \( t=0 \): \[ h(0) = -16(0)^2 + 38(0) + 2 = 2 \] So, the intercept is \( (0, 2) \).
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To find the other intercepts (x-intercepts), set \( h(t) = 0 \): \[ -16t^2 + 38t + 2 = 0 \] Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ t = \frac{-38 \pm \sqrt{38^2 - 4 \cdot (-16) \cdot 2}}{2 \cdot (-16)} = \frac{-38 \pm \sqrt{1444 + 128}}{-32} = \frac{-38 \pm \sqrt{1572}}{-32} \] Calculating this gives us two values. The relevant positive root is \( t \approx 2.38 \) seconds.
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Finding the vertex:
- The time to reach the maximum height (vertex) occurs at \( t = -\frac{b}{2a} \): \[ t = -\frac{38}{2 \cdot -16} = \frac{38}{32} = 1.19 \text{ seconds} \]
- Now, substituting \( t=1.19 \) to find the maximum height: \[ h(1.19) = -16(1.19)^2 + 38(1.19) + 2 \approx 24.56 \text{ feet} \]
- Thus, the vertex is \( (1.19, 24.56) \).
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Path of the shirts: After being launched from the cannon at a height of 2 feet, the shirts reach a maximum height of approximately 24.56 feet after about 1.19 seconds before returning to ground level at 2.38 seconds.
Conclusion: The correct response from the options provided is:
(0,2), (1.19,24.56), and (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.
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