1. **State the problem:** We have a piecewise function for filing income taxes $T(x)$ based on adjusted gross income $x$: $$ T(x) = \begin{cases} 0.15x & 0 \leq x \leq 18000 \\ 0.28(x - 18000) + 2700 & 18000 < x \leq 44250 \\ 0.32(x - 44250) + 10050 & x > 44250 \end{cases} $$ We need to find and interpret $T(40000)$ and $T(75000)$. 2. **Find $T(40000)$:** Since $40000$ is between $18000$ and $44250$, use the second piece: $$ T(40000) = 0.28(40000 - 18000) + 2700 $$ Calculate inside the parentheses: $$ 40000 - 18000 = 22000 $$ Multiply: $$ 0.28 \times 22000 = 6160 $$ Add 2700: $$ T(40000) = 6160 + 2700 = 8860 $$ Interpretation: The tax on $40000$ income is 8860 dollars. 3. **Find $T(75000)$:** Since $75000 > 44250$, use the third piece: $$ T(75000) = 0.32(75000 - 44250) + 10050 $$ Calculate inside the parentheses: $$ 75000 - 44250 = 30750 $$ Multiply: $$ 0.32 \times 30750 = 9840 $$ Add 10050: $$ T(75000) = 9840 + 10050 = 19890 $$ Interpretation: The tax on $75000$ income is 19890 dollars. **Final answers:** - $T(40000) = 8860$ - $T(75000) = 19890$