Problem

A population starts at \(P_{0}~>~0\) and grows by approximately the same positive \(r\%\) each year. Analyze the difference and ratio behavior, determine the model family and general model, interpret the growth factor, cite decisive evidence, and explain why the other family is inappropriate.

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differences = not constant for equal percent growth; ratios = approximately 1+r/100, constant; family = exponential; model = P(t)=P₀(1+r/100)ᵗ; change = r%/year; reject linear = equal percentages do not give equal additions

differences = r/100 each year, constant; ratios = not needed; family = linear; model = P(t)=P₀+(r/100)t; change = r/100 people/year; reject exponential = percentages are additive

differences = grow each year; ratios = not relevant; family = quadratic; model = P(t)=P₀+(r/100)t²; change = increasing additive rate; reject exponential = growth cannot be multiplicative

differences = not constant; ratios = r/100 each year; family = exponential; model = P(t)=P₀(r/100)ᵗ; change = retains r%/year; reject linear = changes are multiplicative

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A population starts at \(P_{0}~>~0\) and grows by approximately the same positive \(r\%\) each year. Analyze the difference and ratio behavior, determine the model family and general model, interpret the growth factor, cite decisive evidence, and explain why the other family is inappropriate.

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