Problem

A population starts at \(P_{0}\) and grows \(6\%\) each year \(t\). Determine the model family, equation and domain, parameter meanings, decisive evidence, and why a linear model is inappropriate.

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family = quadratic; model = P(t)=P₀+0.06t² for t≥0; parameters = P₀ initial population and 0.06 people/year squared; evidence = annual increases get larger; reject linear = growth cannot have constant rate

family = exponential; model = P(t)=P₀(1.06)ᵗ for t≥0; parameters = P₀ initial population and 1.06 yearly factor; evidence = each year multiplies by 1.06; reject linear = 6% of changing population is not constant addition

family = linear; model = P(t)=P₀+0.06t for t≥0; parameters = P₀ initial population and 0.06 people/year; evidence = 6% means add 0.06 each year; reject exponential = population cannot multiply

family = exponential; model = P(t)=P₀(0.06)ᵗ for t≥0; parameters = P₀ initial population and 0.06 yearly factor; evidence = a 6% change uses factor 0.06; reject linear = yearly change is not additive

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A population starts at \(P_{0}\) and grows \(6\%\) each year \(t\). Determine the model family, equation and domain, parameter meanings, decisive evidence, and why a linear model is inappropriate.

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