Problem

For \(C(t)~=~t(t~-~5)/[(t~-~5)(t~+~1)]\) with \(t~\ge~0~\text{seconds}\), cancel valid factors, retain every original restriction, state the reduced model, locate the hole, identify the remaining vertical asymptote, and compare \(\text{both}~\text{exclusions}\) with the contextual domain.

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Cancellation: remove t−5 but retain t≠5 and t≠−1. Reduced model: C=t/(t+1) for t≥0, t≠5. Hole: (5,0). Remaining vertical asymptote: t=−1, outside the contextual domain.

Cancellation: remove t−5 and restore t=5, retaining only t≠−1. Reduced model: C=t/(t+1) for all t≥0. Hole: none. Remaining vertical asymptote: t=−1, outside the contextual domain.

Cancellation: remove both denominator factors. Reduced model: C=t for t≥0, t≠5. Holes: t=5 and t=−1. Remaining vertical asymptote: none.

Cancellation: remove t−5 but retain t≠5 and t≠−1. Reduced model: C=t/(t+1) for t≥0, t≠5. Hole: (5,5/6). Remaining vertical asymptote: t=−1, outside the contextual domain.

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For \(C(t)~=~t(t~-~5)/[(t~-~5)(t~+~1)]\) with \(t~\ge~0~\text{seconds}\), cancel valid factors, retain every original restriction, state the reduced model, locate the hole, identify the remaining vertical asymptote, and compare \(\text{both}~\text{exclusions}\) with the contextual domain.

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