The city of Arlington is determining its tax rate structure for the coming year. The city needs to generate $$6$ million in tax revenue via taxes of property, sales, prepared food, and utilities. The following table summerizes how much tax revenue would be generated from each segment of the population by the $1\%$ increase in each tax category. $($For instance. a $2\%$ tax on prepared food would generate $$240,000$ in tax revenue from upper income residents.$)$

Revenues $($in $$1,000$s$)$ per $1\%$ Tax Rate.

Income Group Sales Property Food Utility

Low $$200$ $$600$ $$50$ $$80$

Middle $$250$ $$800$ $$100$ $$110$

Upper $$400$ $$1200$ $$130$ $$120$

City commissioners have specified that the tax rate for each revenue category must be between $1\%$ and $3\%$ and the tax rate on prepared food cannot exceed half the sales tax rate. Ideally, the commissioners have a goal of making up the $$6$ million tax budget with $$1\mathrm{.}5$million from low income residents, $$2\mathrm{.}1$ million from middle income residents, and $$2\mathrm{.}3$ million from high income residents. If that is not possible, the commissioners would like a solution that minimizes the maximum percentage deviation from these tax revenue goals for each income group.

a$.$ Create a spreadsheet model for this problem.

b$.$ What is the optimal solution?