Problem 116 - College Algebra Essentials

Question

The bar graph shows the total amount Americans paid in federal taxes, in trillions of dollars, and the U.S. population, in millions, from 2007 through 2010. CAN'T COPY THE GRAPH a. In \(2009,\) United States government collected \(\$ 2.20\) trillion in taxes. Express this number in scientific notation. b. In \(2009,\) the population of the United States was approximately 308 million. Express this number in scientific notation. c. Use your scientific notation answers from parts (a) and (b) to answer this question: If the total 2009 tax collections were evenly divided among all Americans, how much would each citizen pay? Express the answer in decimal notation, rounded to the nearest dollar.

Step-by-Step Solution

Answer

a. The number \$2.20 trillion in scientific notation is \(2.20 \times 10^{12}\). b. The number 308 million in scientific notation is \(3.08 \times 10^{8}\). c. If the total 2009 tax collections were evenly divided among all Americans, each citizen would pay approximately \$7143.

Step 1: Part a: Convert taxes into scientific notation

Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form. To express the number \(\$2.20\) trillion in scientific notation, we first write it as \(2.20 \times 10^{12}\) since one trillion equals \(10^{12}\). The scientific notation is then \(2.20 \times 10^{12}\).

Step 2: Part b: Convert population into scientific notation

To express the number 308 million in scientific notation, we first write it as \(308 \times 10^{6}\) since one million equals \(10^{6}\). The scientific notation is then \(3.08 \times 10^{8}\).

Step 3: Part c: Calculate tax per citizen

The total tax collections divided by the total population gives the tax per citizen. Thus, we divide \(2.20 \times 10^{12}\) by \(3.08 \times 10^{8}\) to get \(7.142857142857143 \times 10^{3}\). Rounded to the nearest dollar, this becomes \$7143 in decimal notation.

Key Concepts

Federal Taxes

Federal taxes refer to the money that the government collects from individuals and businesses to fund public services and operations. In 2009, the United States government collected \$2.20 trillion in federal taxes. This hefty sum is used to support various federal programs such as:

• National defense
• Social Security and healthcare
• Infrastructure projects

Often, large financial figures like these are expressed in scientific notation, a method that simplifies unwieldy numbers by using powers of ten. So, \(2.20 \text{ trillion}\) becomes \(2.20 \times 10^{12}\). This method makes complex calculations and comparisons more manageable, especially when dealing with national data like taxes.

Population

Population figures refer to the total number of people residing in a certain area, such as a country or city. In 2009, the population of the United States was approximately 308 million people. When handling such large numbers, scientific notation offers a simplified way to express them, which is very useful in statistics and data representation.

For example, 308 million can be written as \(3.08 \times 10^{8}\). This notation allows for easier calculation and understanding, particularly when working with other large numbers. It also aids in dividing or multiplying quantities when assessing per capita statistics, such as the average tax burden per citizen.

Decimal Notation

Decimal notation is a standard way of writing numbers using the decimal system, based on powers of ten. It is what we commonly use in everyday arithmetic. After calculating a value using scientific notation or other methods, the final answer is often converted back into decimal notation to make it more understandable.

In the context of dividing total federal taxes by the population number, once you have the scientific notation result \(7.142857142857143 \times 10^{3}\), converting it into decimal notation gives you 7143. This is because \(7.142857142857143 \times 10^{3}\) is the same as 7,143 in decimal form. Rounding this figure to the nearest dollar confirms that, if the total tax amount were distributed evenly among all Americans, each would pay approximately \$7,143.