Problem

Population is modeled by \(P(t)~=~500\cdot~1.08^{t}~\text{people}\) \(t~\text{years}\) after observation begins. Interpret the input and output, initial value, yearly growth factor and percent change, and consecutive-output ratio.

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What this problem is really about

An exponential model compresses several contextual ideas into its coefficient, base, and exponent. We’ll first attach the input and output to their real-world units, then use a zero input to expose the starting amount. Finally, we’ll separate the multiplicative factor from the percent increase and connect that factor to the ratio of consecutive outputs.

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input = t years; output = P(t) people; initial = P(0)=500 people; factor = 1.08; change = 108% growth per year; ratio meaning = P(t+1)/P(t)=2.08

input = t people; output = P(t) years; initial = P(0)=1.08 people; factor = 500; change = 500% growth per year; ratio meaning = P(t+1)/P(t)=500 years per person

input = t years; output = P(t) people; initial = P(0)=500 people; factor = 1.08; change = 8% growth per year; ratio meaning = P(t+1)/P(t)=1.08, so each year's modeled population is 108% of the previous year's

input = t years; output = P(t) people; initial = P(0)=500 people; factor = 1.08 people per year; change = 8 people gained per year; ratio meaning = P(t+1)-P(t)=1.08

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Population is modeled by \(P(t)~=~500\cdot~1.08^{t}~\text{people}\) \(t~\text{years}\) after observation begins. Interpret the input and output, initial value, yearly growth factor and percent change, and consecutive-output ratio.

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