jmc

If Dr. Jones has an income of $x$, then the amount of tax is essentially a piecewise function in $x$. In particular, if we let $t(x)$ denote the amount of taxes, then $t(x)=0$ when $0\le x\le 20000$. For $20000\le x\le 45000$, he pays $$t(x)=0.05(x-20000).$$For $45000\le x\le 80000$, he pays $$\begin{array}{rl}t(x)& =0.05(45000-20000)+0.1(x-45000)\\ & =1250+x/10-4500.\end{array}$$For $80000\le x\le 130000$ he pays $$\begin{array}{rl}t(x)& =1250+0.1(80000-45000)+0.15(x-80000)\\ & =4750+0.15x-12000.\end{array}$$Finally, if $x\ge 130000$, he pays $$\begin{array}{rl}t(x)& =4750+0.15(130000-80000)+0.2(x-130000)\\ & =12250+0.2(x-130000).\end{array}$$We can immediately eliminate the last possibility, since then he would automatically pay at least \12,250$intaxes.If$x \le 80000$,then$t(x) \le 1250 + 80000/10 - 4500 = 4750$.Hence,$80000 \le x \le 130000$. Then, $$10000=4750+0.15x-12000⟹x=\boxed{\$115,000}.$$