New York state income tax is based on taxable income, which is part of a person's total income. The tax owed to the state is calculated using the taxable income $($not total income$).$ In $2016,$ for a single person with a taxable income between $$80,150$ and $$214,000,$ the tax owed was $$4833$ plus $6\mathrm{.}65\%$ of the taxable income over $$80,{150}^1.$

$($a$)$ Compute the tax owed by a lawyer whose taxable income is $$99,000.$ NOTE: Round to the nearest cent.

Tax owed $=$

$($b$)$ Consider a lawyer whose taxable income is $90\%$ of her total income, $x,$ where $x$ is between $$110,000$ and $$200,000.$ Assuming her total income is between $$110,000$ and $$200,000,$ write a

formula for $T(x),$ the taxable income.

$T(x)=$

$($c$)$ Write a formula for $L(x),$ the amount of tax owed by the lawyer in part $($b$).$ NOTE: Enter the exact answer or round to $3$ decimal places.

$L(x)=$

$($d$)$ Use $L(x)$ to evaluate the tax owed for $x=110,000$ and compare your results to part $($a$).$ NOTE: Round to the nearest cent.

Tax owed $=$$

The values from $($a$)$ and $($d$)$ are