Problem

A quantity is \(200~\text{units}\) at month \(m~=~0\) and decreases by \(25\%\) each month. Determine whether the pattern is linear or exponential. Give the model and domain, interpret its parameters, calculate the next two values, cite decisive evidence, and explain why the other family is inappropriate.

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family = exponential; model = A(m)=200·0.25ᵐ for m∈0,1,2,…; parameters = initial 200 units and retained factor 0.25; values = A(1)=50 and A(2)=12.5; evidence = 25% decrease retains 25%; reject linear = changes are multiplicative

family = exponential; model = A(m)=200·1.25ᵐ for m∈0,1,2,…; parameters = initial 200 units and factor 1.25; values = A(1)=250 and A(2)=312.5; evidence = 25% change uses factor 1.25; reject linear = values change by percent

family = linear; model = A(m)=200-25m for m∈0,1,2,…; parameters = initial 200 units and rate -25 units/month; values = A(1)=175 and A(2)=150; evidence = 25% means subtract 25 units; reject exponential = percentages are additive

family = exponential; model = A(m)=200·0.75ᵐ for m∈0,1,2,…; parameters = initial 200 units, retained factor 0.75, decay 25%/month; values = A(1)=150 and A(2)=112.5; evidence = each output multiplies by 0.75; reject linear = absolute decreases are not constant

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