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Sierra King is a nail technician. She allots 20 minutes for a manicure and 45 minutes for a pedicure in her 7-hour work day. No more than 5 pedicures can be scheduled each day. The prices are $18 for a manicure and $45 for a pedicure. How many manicures and pedicures should Ms. King schedule to maximize her daily income? What is her maximum daily income?



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luana
[tex]x-\ number\ of\ manicure\\y-\ number\ of\ pedicure\\\\ 20min=\frac{1}{3}hour\\ 45min=\frac{3}{4}hour\\\\ y \leq 5\\\\ \frac{1}{3}*x+\frac{3}{4}y=7\ \ \ |multiply\ by\ 12\\ 4x+9y=84\\\\More\ cost-effective \ for\ her\ will\ be\ to\ have\ only\ pedicures\\ because\ making\ manicure\ for\ 20 min\ she\ has\ 18\$\\ so\ for\ 1\ minute\ it\ is\ 0,9\$\\while\ making\ pedicure\ she\ has\ 45\$\ for\ 45\ minutes\ so\\ it\ is\ 1\$\ for\ 1\ minute\ of\ work\\\\ But\ she\ can\ make\ 5\ pedicures\ maximally[/tex][tex]so\ we\ take\ y=5 4x+9*5=84\\\\ 4x+45=84\ \ \ \ | subtract\ 45\\\\ 4x=39\ \ \ |divide\ by\ 4\\\\x=9,75-\ she\ is\ not\ able\ to\ make\ 10\ manicures\\ so\ we\ will\ change\ number\ of\ pedicures\ to\ y=4\\\\ 4x+9*4=84\\ 4x+36=84\\\\ 4x=48\ \ \ |divide\ by\ 4\\\\ x=12\\\\ She\ should\ make\ \boxed{12\ manicures\ and\ 4\ pedicures.}\\\\ Income= 12*18\$+4*45\$=216+180=\boxed{396\$}[/tex]

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