Problem

A ball launched at \(52~\text{feet}~\text{per}~\text{second}\) from \(12~\text{feet}\) has height \(h(t)=-16t^{2}+52t+12\). Find every time before landing when the ball is \(48~\text{feet}~\text{high}\). Reduce \(h(t)=48\) to zero, factor it, check each root against the landing domain, and verify both retained times in the model.

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Set h(t) = 48, but no height model or algebraic reduction is supplied, so no time can be determined.

The solutions include t = −1, 1, and 9/4 seconds. The factorization gives 1 and 9/4, and negative time is outside the context.

h(t) = −16t² + 52t + 12. Setting h(t) = 48 gives 4t² − 13t + 9 = (4t − 9)(t − 1) = 0, so t = 1 and 9/4 seconds; both are before landing and give 48 feet.

The target 48 feet is treated as input t = 48 seconds. This is outside the landing domain and confuses output with input.

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