Problem 76 - Algebra and Trigonometry

Question

Income Tax In a certain country, income tax T is assessed according to the following function of income \(T(x)=\left\\{\begin{array}{ll}{0} & {\text { if } 0 \leq x \leq 10,000} \\\ {0.08 x} & {\text { if } 10,000

Step-by-Step Solution

Answer

Taxes are: $0 at $5,000, $960 at $12,000, $5350 at $25,000. Results represent tax amounts for respective incomes.

Step 1: Determine Tax for $5000

For an income of \(5000, we apply the first condition of the tax function, as \)5000 falls within the range of \(0 \leq x \leq 10,000\). In this range, the tax \(T(x)\) is \(0\). Therefore, \(T(5000) = 0\).

Step 2: Determine Tax for $12000

For an income of \(12000, apply the second condition, since \)12000 is within the interval \(10,000 < x \leq 20,000\). According to this condition, the tax \(T(x) = 0.08x\). So, \(T(12000) = 0.08 \times 12000 = 960\).

Step 3: Determine Tax for $25000

For an income of \(25000, apply the third condition since \)25000 falls above $20,000. Here, the tax function is \(T(x) = 1600 + 0.15x\). Thus, \(T(25000) = 1600 + 0.15 \times 25000 = 1600 + 3750 = 5350\).

Step 4: Interpretation of Results

The results from part (a) represent the amount of income tax a person would pay based on their income levels. Specifically, income below $10,000 incurs no tax, income at $12,000 incurs a tax of $960, and income at $25,000 results in a tax of $5350.

Key Concepts

Income Tax Calculation

Income tax calculation is an important aspect of financial planning. It determines how much money individuals owe to the government based on their income. This is typically done through a piecewise function, which calculates tax differently over predefined income intervals. For our exercise, the tax function is defined for three income ranges:

• Income between \(0 and \)10,000 incurs no tax (\[ T(x) = 0 \]).
• Income between \(10,000 and \)20,000 is taxed at 8% (\[ T(x) = 0.08x \]).
• Income above \(20,000 is taxed at 15% with a base amount of \)1,600 (\[ T(x) = 1600 + 0.15x \]).

Using mathematical expressions and conditions to determine tax ensures individuals in different income brackets pay fair and proportionate taxes.

Tax Brackets

Tax brackets divide income into levels, each with different tax rates. Understanding them is crucial, as they determine how much tax you pay. The function given in our exercise assigns different rates to income tiers:

• Up to $10,000: The tax rate is 0%, meaning you keep all your income.
• $10,001 to $20,000: Tax is calculated at a rate of 8%.
• More than $20,000: Here, a base tax of $1,600 is applied, plus 15% on the income exceeding $20,000.

Tax brackets ensure that the tax system is progressive, meaning higher earnings are taxed at higher rates. This approach is designed to be equitable and cater to varying financial capabilities of earners.

Function Evaluation

Function evaluation involves finding specific outputs from given inputs within a mathematical function. In our tax function problem, it means substituting income values into the piecewise function to calculate the exact tax amount. For instance, for an income of \(5,000, no tax is payable as it falls in the first bracket. However, for \)12,000, plugging in the value: \[ T(12000) = 0.08 \times 12000 = 960 \] shows that the tax is \(960, applying the second bracket's rules. Similarly, evaluating for \)25,000: \[ T(25000) = 1600 + 0.15 \times 25000 = 5350 \] indicates a tax of $5,350, following the third bracket's conditions. Through function evaluation, we find accurate tax amounts depending on where the income lies within the defined tax brackets. This systematic method helps apply uniform rules for calculating taxes, ensuring consistency across different income levels.