Problem

Compare service costs $A(h)=15h+20$ and $B(h)=12h+35$ by giving their starts and hourly rates, solving for the crossover and cost there, and identifying the lower-cost model before and after it.

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What this problem is really about

Comparing two linear costs requires treating starting fees and hourly rates as separate advantages. We’ll compare both parameter pairs, set the models equal to locate their crossover, and verify the common cost there. The difference in slopes will then determine which model is lower before the meeting point and which is lower afterward.

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starts = A=$20, B=$35; rates = A $15/hour, B $12/hour; crossover = h=15; cost comparison = A cheaper before 15

starts = A(0)=$20, B(0)=$35; rates = A $15/hour, B $12/hour; crossover = 15h+20=12h+35 ⇒ h=5, cost $95; cost comparison = A costs less for h<5, equal at 5, B costs less for h>5

starts = A=$20, B=$35; rates = A $15/hour, B $12/hour; crossover = none; cost comparison = A always cheaper because it starts lower

starts = A=$20, B=$35; rates = A $15/hour, B $12/hour; crossover = h=5; cost comparison = B costs less before 5 and A after 5

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Compare service costs \(A(h)=15h+20\) and \(B(h)=12h+35\) by giving their starts and hourly rates, solving for the crossover and cost there, and identifying the lower-cost model before and after it.

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