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Chapter 1: Problem 1

Tax owed: The income tax \(T\) owed in a certain state is a function of the taxable income \(I\), both measured in dollars. The formula is \(T=0.11 I-500\) a. Express using functional notation the tax owed on a taxable income of \(\$ 13,000\), and then calculate that value. b. If your taxable income increases from \(\$ 13,000\) to \(\$ 14,000\), by how much does your tax increase? c. If your taxable income increases from \(\$ 14,000\) to \(\$ 15,000\), by how much does your tax increase?

Short Answer

Expert verified
a) $930; b) $110; c) $110.

Step by step solution

01

Determine Function Notation for Tax

We are given the formula for tax as \( T = 0.11I - 500 \). To express the tax owed on a taxable income of \( \$13,000 \) using functional notation, we need to substitute \( I = 13,000 \) into the equation. This gives us: \( T(13,000) = 0.11(13,000) - 500 \).
02

Calculate Tax for $13,000 Income

Substitute \( I = 13,000 \) into the formula: \( T(13,000) = 0.11 \times 13,000 - 500 \). Calculate the result: \( 0.11 \times 13,000 = 1,430 \). Then subtract 500: \( 1,430 - 500 = 930 \). Therefore, \( T(13,000) = 930 \).
03

Calculate Tax for $14,000 Income

Find the tax owed when \( I = 14,000 \) using the formula: \( T(14,000) = 0.11(14,000) - 500 \). Calculate: \( 0.11 \times 14,000 = 1,540 \). Subtract 500: \( 1,540 - 500 = 1,040 \). Hence, \( T(14,000) = 1,040 \).
04

Determine Tax Increase from $13,000 to $14,000

Find the difference in tax between \( \\(14,000 \) and \( \\)13,000 \) incomes: \( T(14,000) - T(13,000) = 1,040 - 930 = 110 \). The tax increase is \( \$110 \).
05

Calculate Tax for $15,000 Income

Determine the tax owed for \( I = 15,000 \) using the formula: \( T(15,000) = 0.11(15,000) - 500 \). Compute: \( 0.11 \times 15,000 = 1,650 \). Subtract 500: \( 1,650 - 500 = 1,150 \). Therefore, \( T(15,000) = 1,150 \).
06

Determine Tax Increase from $14,000 to $15,000

Calculate the difference in tax from incomes \( \\(14,000 \) to \( \\)15,000 \): \( T(15,000) - T(14,000) = 1,150 - 1,040 = 110 \). The tax increases by \( \$110 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taxable Income
Taxable income is a key concept in determining how much tax a person owes. It essentially refers to the portion of a person's income that is subject to taxation. Consider it as the income you earn minus any deductions or exemptions that the government allows to be taken into account to reduce your total income.

To calculate the tax owed based on taxable income, a formula is applied. For instance, in this exercise, the formula given is:
  • \( T = 0.11I - 500 \)
where \( T \) represents the tax owed, and \( I \) represents the taxable income.

This formula has a linear component \( 0.11I \) that indicates a portion of the taxable income is multiplied by a fixed rate (11%), showing that as your income increases, your taxes increase by a constant rate, minus a fixed amount like the \( 500 \) dollars represented here.
Functional Notation
Functional notation is a convenient way to represent how one quantity depends on another. In algebra, functions are used to describe relationships between variables. For example, the tax owed function can be expressed as \( T(I) \), where \( T \) is a function of taxable income \( I \).

When you see a notation like \( T(13,000) \), it implies that you are evaluating the function \( T \) at a specific taxable income of \( 13,000 \). This helps visualize and compute the corresponding tax amount through simple substitution in the formula provided. Thus, the expression \( T(13,000) = 0.11 \times 13,000 - 500 \) offers a clear, functional view on how the tax is calculated for this income level.

In broader terms, functional notation provides a method to examine variable relationships by specifying inputs and observing the produced output, making it easier to manage and solve problems in algebra.
Rate of Change
The rate of change concept reveals how one quantity changes in relation to another. In terms of income tax, it provides insight into how tax amounts change as taxable income changes.

In the exercise, when taxable income increases from \( 13,000 \) to \( 14,000 \), the tax increases by \( 110 \). Similarly, from \( 14,000 \) to \( 15,000 \), there is also a \( 110 \) increase.
  • This consistent increase implies a fixed rate of change, reflecting the linear nature of the relationship between income and tax in this scenario.
A rate of change could be referred to as the slope in a line equation, indicating how steep or gradual changes occur as one variable influences another. The formula \( T = 0.11I - 500 \) captures this rate, with the coefficient \( 0.11 \) representing the constant change in tax per unit of income.

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Most popular questions from this chapter

Yellowfin tuna: Data were collected comparing the weight \(W\), in pounds, of a yellowfin tuna to its length \(L\), in centimeters. \({ }^{19}\) These data are presented in the table below. $$ \begin{array}{|c|c|} \hline L=\text { Length } & W=\text { Weight } \\ \hline 70 & 14.3 \\ \hline 80 & 21.5 \\ \hline 90 & 30.8 \\ \hline 100 & 42.5 \\ \hline 110 & 56.8 \\ \hline 120 & 74.1 \\ \hline 130 & 94.7 \\ \hline 140 & 119 \\ \hline 160 & 179 \\ \hline 180 & 256 \\ \hline \end{array} $$ a. What is the average rate of change, in weight per centimeter of length, in going from a length of 100 centimeters to a length of 110 centimeters? b. What is the average rate of change, in weight per centimeter, in going from 160 to 180 centimeters? c. Judging from the data in the table, does an extra centimeter of length make more difference in weight for a small tuna or for a large tuna? d. Use the average rate of change to estimate the weight of a yellowfin tuna that is 167 centimeters long. e. What is the average rate of change, in length per pound of weight, in going from a weight of 179 pounds to a weight of 256 pounds? f. What would you expect to be the length of a yellowfin tuna weighing 225 pounds?

Parallax: If we view a star now, and then view it again 6 months later, our position will have changed by the diameter of the Earth's orbit around the sun. For nearby stars (within 100 light-years or so), the change in viewing location is sufficient to make the star appear to be in a slightly different location in the sky. Half of the angle from one location to the next is known as the parallax angle (see Figure 1.5). Parallax can be used to measure the distance to the star. An approximate relationship is given by $$ d=\frac{3.26}{p}, $$ where \(d\) is the distance in light-years, and \(p\) is the parallax measured in seconds of arc. \({ }^{5}\) Alpha Centauri is the star nearest to the sun, and it has a parallax angle of \(0.751\) second. How far is Alpha Centauri from the sun? Note: Parallax is used not only to measure stellar distances. Our binocular vision actually provides the brain with a parallax angle that it uses to estimate distances to objects we see.

Altitude: A helicopter takes off from the roof of a building that is 200 feet above the ground. The altitude of the helicopter increases by 150 feet each minute. a. Use a formula to express the altitude of a helicopter as a function of time. Be sure to explain the meaning of the letters you choose and the units. b. Express using functional notation the altitude of the helicopter 90 seconds after takeoff, and then calculate that value. c. Make a graph of altitude versus time covering the first 3 minutes of the flight. Explain how the description of the function is reflected in the shape of the graph.

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