Chapter 1.5, Problem 27E

Computing Income Tax The tax that you pay to the federal government is a percentage of your taxable income, which is what remains of your gross income after you subtract your allowed deduction. In a recent year, there were five rates or brackets for a single taxpayer as shown in Table 1

$\begin{array}{l}\text{Single}\text{Taxpayer}\text{Rates}\\ \begin{array}{ccc}\text{Amount}\text{Over}& \text{But}\text{Not}\text{Over}& \text{Tax}\text{Rate}\\ \$0& \$27,050& 15\%\\ \$27,050& \$65,550& 27.5\%\\ \$65,550& \$136,750& 30.5\%\\ \$136,750& \$297,350& 35.5\%\\ \$297,350& \mathrm{...}& 39.1\%\end{array}\end{array}$

So, if you are single and your taxable income was less that $\$27,050$, your tax is your taxable income times $15\%(0.15)$. The maximum amount of tax that you will pay on your income in the first bracket is $15\%$ of $\$27,050$ or $(0.15)\times 27,050=4057.50$ dollars. If your income is more than $\$27,050$ but less than $\$65,550$ your tax is $\$4057.50$ plus $27.5\%$ of the amount in excess of $\$27,050$. So, for example, if your taxable income is $\$50,000$, your tax is

$4057.5+.275(50,000-27,050)=4057.5+.275\times 22950=\$10,368.75$

Let $x$ denote your taxable income and $T(x)$

Find the formula for the $T(x)$ if $x$ is not over $\$136,750$.

Plot the graph of $T(x)$ for $T(x)$ for $0\le x\le 136,750$.

Find the maximum amount of tax that you will pay on the portion of your income in second tax bracket. Expression this as a difference between two values of $T(x)$