Problem

Interpret \(T(t)=20+60\cdot~0.5^{t}\) for \(t\ge~0\) by identifying the baseline, initial excess and total, retention and decay rates, approach behavior, and range.

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What this problem is really about

A shifted cooling model separates the surrounding baseline from the temperature excess above it. We’ll evaluate the initial excess and total, interpret the decay factor as acting only on that excess, and track its positive approach toward zero. That structure determines both the horizontal asymptote and which endpoint is excluded from the finite-time range.

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baseline/asymptote = 20°C; initial excess = 60°C; total initial = 80°C; excess factor = entire temperature halves; approach = toward 0°C; range = [0,80]

baseline/asymptote = 20°C; initial excess = 60°C above baseline; total initial = T(0)=80°C; excess factor = 0.5, retains 50%/decreases 50% each interval; approach = decreases toward 20°C from above; range for t≥0 = (20,80]

baseline/asymptote = 20°C; initial excess = 60°C; total initial = 60°C; excess factor = 0.5; approach = toward 20°C; range = (20,60]

baseline/asymptote = 20°C; initial excess = 60°C; total initial = 80°C; excess factor = 0.5; approach = reaches 20°C at finite time; range = [20,80]

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Interpret \(T(t)=20+60\cdot~0.5^{t}\) for \(t\ge~0\) by identifying the baseline, initial excess and total, retention and decay rates, approach behavior, and range.

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