$01$Question$(8$ points$)$

A growing metropolis has authorized the expansion of commuter trains to a broader region around the city. The city government conducted a competitive bidding process and has awarded the contract for the new train lines to Acme train company.$$

Acme's base profits for this venture are represented by$=962$T$=96$x$$x$2,$ where$$x$$represents the number of new train lines established. However, the new train lines will run adjacent to land owned by a developer, who plans to build new houses on the land. Consumers generally do not want to live adjacent to noisy train tracks, so the developer's profits are impacted by the train lines. The profits for the developer are$=842$D$=84$y$$y$2$xy$,$ where$$y$$is the number of houses she builds.$$

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$3$rd attempt

Part $1(4$pts$)$

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After considering the impact of the train lines on housing, the city government has determined that the train company must compensate the developer for any loss in property value from the trains. Now, for every train line that is run near the developer's land, the train company must pay$$y$$to the developer.$$

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How many train lines will the train company run?$$ What will its total profits be$?$$$$

How many houses will the developer build?$$ What will her total profits be$?$$$$