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You constructed a piecewise-defined function from the $2000$Federal Tax Rate Schedule that you will use in the next two problems. Specifically, you found that a person who makes m dollars a year will pay $T\left(m\right)$dollars in tax, given by the function

$\begin{array}{r}0.15m,\text{if}0≤m≤26,250\\ 3,937+0.28(m−26,250),\text{if}26,250<m≤63,550\\ 14,381+0.31(m−63,550),\text{if}63,550<m≤132,600\\ 35,787+0.36(m−132,600),\text{if}132,600<m≤288,350\\ 91,857+0.396(m−288,350),\text{if}m>288,350\end{array}$

Suppose you make role="math" localid="1648136183164" $\$63,550$a year and pay tax according to the given formula.

(a) Calculate the value of T(63,550) and the limit of $T\left(m\right)$as m approaches 63,550 from the left and from the right.

(b) Use part $\left(a\right)$to argue that the function $T\left(m\right)$is continuous at $m=63,550$. What does this mean in real-world terms?

Step by step solution

01

Step 1. Given information.

given,$m=63550$

02

Step 2. Put m=63550 in the function to get the value of T(63550)as below:

$\begin{array}{r}T\left(m\right)=3937+0.28(m−26250)\\ T\left(63550\right)=3937+0.28(63550−26250)\\ =3937+0.28\left(37300\right)\\ =3937+10444\\ =14381\end{array}$

03

Step 3. Use part (a) to argue that the function T(m) is continuous at m=63,550.

The value for $T\left(63550\right)$ matches up with the value of the next piece $14381+0.31(m-63550)$when $m=63550$. This means that when you change tax brackets, the higher tax rate only applies to the amount over $63550$ .

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