Optimizing Tax Brackets

00:00 6 marks

Consider a progressive income tax system where the tax rate is defined by a piecewise function. For an income \(x\) (in dollars), the tax payable \(T(x)\) is given by:

– If \(0 \le x < 30000\): \(T(x) = 0.1x\)
– If \(30000 \le x < 100000\): \(T(x) = 3000 + 0.2(x – 30000)\)
– If \(x \ge 100000\): \(T(x) = 15000 + 0.3(x – 100000)\)

Answer the following:

(a) Identify the income intervals corresponding to each tax bracket.

(b) Determine the tax paid by an individual earning exactly \(30000\) dollars.

Part a

3 minutes 3 marks

Question

Identify the income intervals corresponding to each bracket based on the tax function:

\(T(x) = \begin{cases}\;0.1x,& \text{if } 0 \leq x < 30000\\\;3000 + 0.2(x-30000),& \text{if } 30000 \leq x < 100000\\\;15000 + 0.3(x-100000),& \text{if } x \geq 100000\end{cases}\)

List each interval clearly.

Hint Hint

Hint

Look at the conditions provided for \(x\) in each piece of the function.

Answer Answer
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Answer Answer

Complete Answer

Interval 1: \(0 \leq x < 30000\); Interval 2: \(30000 \leq x < 100000\); Interval 3: \(x \geq 100000\)

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Part b

7 minutes 3 marks

Question

Using the intervals identified, determine the tax paid by an individual earning exactly \(30000\) dollars.

Hint Hint

Hint

Substitute \(x = 30000\) into the appropriate piece of the tax function, considering which interval includes \(x = 30000\).

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

The tax paid is \(3000\) dollars.

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