Problem

Use the displayed graphs to analyze \(T(t)=60(0.8)^t+20\). Separate the additive baseline from the exponential component, evaluate \(T(0)\), describe the component’s decay, and identify the long-run temperature and horizontal asymptote.

Unannotated graph of T(t)=60(0.8)^t+20 degrees Celsius; the curve begins at 80 and approaches 20, but baseline and component labels are withheld.

Big Picture

What this problem is really about

A shifted exponential is the sum of a changing component and a fixed baseline. We’ll separate those pieces, evaluate the initial total with the zero-exponent rule, use the base to determine the component’s long-run behavior, and identify the horizontal asymptote as the value left when that component fades away.

Answer choices

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Preview only

Baseline 20°C per interval; component 60(0.8)t°C; T(0) = 80°C; 20°C is added every interval; the long-run temperature increases without bound.

Baseline 80°C; component 60(0.8)t°C; T(0) = 80°C; the component decays toward 0; the long-run temperature and horizontal asymptote are 80°C.

Baseline 20°C; component 60(0.8)t°C; T(0) = 80°C; the component grows without bound; the long-run temperature is 20°C.

Baseline 20°C; component 60(0.8)t°C; T(0) = 80°C; the component decays toward 0; the long-run temperature and horizontal asymptote are 20°C.

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Use the displayed graphs to analyze \(T(t)=60(0.8)^t+20\). Separate the additive baseline from the exponential component, evaluate \(T(0)\), describe the component’s decay, and identify the long-run temperature and horizontal asymptote.

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