Problem

Compare $P(t)=1000\cdot~0.9^{t}$ and $Q(t)=800\cdot~0.95^{t}$ by giving their starts and yearly decay rates, solving for the crossover time, and identifying the greater value before and after it.

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starts = P=$1000, Q=$800; decay = P 10%, Q 5%; crossover = t=40 years from 200/5; comparison = P before, Q after

starts = P=$1000, Q=$800; decay = P 10%, Q 5%; crossover = none; comparison = Q decays faster because 0.95>0.9

starts = P(0)=$1000, Q(0)=$800; decay = P 10%/year, Q 5%/year; crossover = (0.95/0.9)ᵗ=1.25 ⇒ t≈4.127 years; comparison = P>Q before crossover, equal there, Q>P after

starts = P=$1000, Q=$800; decay = P 10%, Q 5%; crossover = none; comparison = P always greater because it starts larger

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Compare \(P(t)=1000\cdot~0.9^{t}\) and \(Q(t)=800\cdot~0.95^{t}\) by giving their starts and yearly decay rates, solving for the crossover time, and identifying the greater value before and after it.

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