Cooper Section $8\mathrm{.}13\backslash$#$9$

A manufacturer wishes to make tee$-$shirts for the band Indigo Girls. They sell for $18$ dollars each. She must deliver all the tee$-$shirts in $20$ days time. The manufacturer will first set up some machines. Once all machines are set up she will turn them on$.$ Each machine takes one day to set up$.$ A machine produces $200$ tee$-$shirts in one day. All the machines must be set before any of them are turned on$.$

a$.$ Express the amount of money she will receive for the shirts in terms of the number of machines $\backslash ($ m $\backslash )$ she decides to set up$.$

$\backslash (\backslash$$ $\backslash )$

b$.$ Use calculus to find out how many machines she should set up to obtain the most money.

$\backslash [$

m$=$

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