Problem

Interpret \(P(t)=500(1.04)^t\) for population \(P\) in people after \(t\) elapsed years. State the units and meaning of \(\text{both}~\text{variables}\), the initial population, the yearly multiplier and percent change, \(P(1)\), and the contextual restriction on \(t\).

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What this problem is really about

Interpreting an exponential model means attaching quantities, units, and intervals to every parameter. We’ll evaluate at zero for the initial output, interpret the unitless base per one input interval, test the next interval, and restrict the input to values meaningful in context.

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P is people and t is elapsed time with no unit; initial population 500; multiplier 1.04 per unspecified interval.

P is people and t is elapsed years; initial population 500; 104% yearly growth; P(1)=1020; t≥0.

P is people and t is elapsed years; population after one year 500; initial population 500/1.04.

P is people and t is elapsed years; initial population 500; 4% yearly growth; P(1)=520; t≥0.

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Interpret \(P(t)=500(1.04)^t\) for population \(P\) in people after \(t\) elapsed years. State the units and meaning of \(\text{both}~\text{variables}\), the initial population, the yearly multiplier and percent change, \(P(1)\), and the contextual restriction on \(t\).

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