Problem

A ball follows \(h=-t^{2}+6t\) for \(t\ge~0\), and a platform is at \(h=8\). Set up and solve the system, preserve both valid times and coordinate order, interpret the intersections, and check the displayed answer graph.

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What this problem is really about

Both relationships describe height, so a valid setup uses the same output variable and equates the ball’s height with the platform’s height. We’ll solve that one-variable quadratic, retain every time allowed by the nonnegative domain, then restore the shared height and interpret each ordered pair in time-height order.

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System: h = −t² + 6t and h = 8. Set equal: −t² + 6t = 8, so t² − 6t + 8 = 0 = (t − 2)(t − 4). Inputs 2 and 4 give points (2, 8) and (4, 8): the ball reaches platform height at 2 and 4 seconds.

System: h = −t² + 6t and h = 8. Set equal: −t² + 6t = 8, so t² − 6t + 8 = 0. Input 2 gives point (2, 8): the ball reaches platform height only once.

System: h = −t² + 6t and h = 8. Set equal: −t² + 6t = 8, so t² − 6t + 8 = 0. Inputs 2 and 4 give points (8, 2) and (8, 4): the ball reaches platform height at those coordinates.

System: h = −t² + 6t and h = 8. Set equal: −t² + 6t + 8 = 0, so t² − 6t − 8 = 0. Inputs are 3 ± √17, giving points (3 − √17, 8) and (3 + √17, 8): these are claimed platform times.

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A ball follows \(h=-t^{2}+6t\) for \(t\ge~0\), and a platform is at \(h=8\). Set up and solve the system, preserve both valid times and coordinate order, interpret the intersections, and check the displayed answer graph.

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