Problem 94 - College Algebra

Question

Use \(10^{12}\) for one trillion and \(2.8 \times 10^{8}\) for the U.S. population in 2000 to solve. In \(2000,\) U.S. personal income was \(\$ 8\) trillion. What was the per capita income, or the income per U.S. citizen? Round to the nearest hundred dollars.

Step-by-Step Solution

Answer

The per capita income, or the income per U.S. citizen in 2000 was approximately $28,600.

Step 1: Understand the problem

In the year 2000, the total personal income in the U.S. was $8 trillion (in scientific notation this is \(8 \times 10^{12}\) dollars) and the total U.S. population was \(2.8 \times 10^{8}\). We are asked to find the per capita income, which is the average income per citizen. This can be calculated by dividing the total personal income by the total population.

Step 2: Perform the division

Divide \(8 \times 10^{12}\) dollars by \(2.8 \times 10^{8}\). Use the property of exponents \(a^m/a^n=a^{m-n}\) to simplify \(10^{12}\) divided by \(10^{8}\) to \(10^4\). The division of 8 by 2.8 gives approximately 2.857.

Step 3: Simplification

Simplify the result to get \(2.857 \times 10^4\) dollars. This is the per capita income in scientific notation.

Step 4: Convert into normal format and round it off

Convert \(2.857 \times 10^4\) dollars into normal format, which is $28,570. Since we are directed to round to the nearest hundred dollars, the average income per citizen becomes approximately $28,600.

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently. This method uses powers of ten to shorten the number. For instance, the number one trillion can be written in scientific notation as \(10^{12}\).
The reason for using scientific notation is to simplify mathematical calculations and express numbers efficiently. For example, instead of writing 1,000,000,000, we can use \(1 \times 10^9\).
When you observe a number in scientific notation like \(8 \times 10^{12}\), it tells you that the number is 8 followed by 12 zeros. This helps when handling numbers like population counts or national income, which can be quite large.
Scientific notation is especially useful when dividing or multiplying large numbers, as it simplifies these operations with the help of exponents.

Division of Exponents

When dividing numbers expressed in scientific notation, one of the essential steps is tackling the exponents. The rule we use is: \(a^m / a^n = a^{m-n}\). This means you simply subtract the exponents.
For instance, if we have \(10^{12}\) and \(10^{8}\), dividing these gives \(10^{12-8} = 10^4\). This simplifies the process significantly, reducing the powers of ten to a manageable expression.
Using this property, our original division \( (8 \times 10^{12}) / (2.8 \times 10^8) \) becomes \((8/2.8) \times 10^4\). This simplifies our calculations by allowing easy handling of large numbers.

Rounding Numbers

Rounding numbers is a technique to make them simpler or fit certain conditions. We often round to a specific place value, such as the nearest hundred to make numbers easier to interpret or relate to.
To round a number, look at the digit right after the place value you are rounding. If it is 5 or higher, you round up. If it's less, you round down.
In our example, after converting \(2.857 \times 10^4\) to the usual format, we get 28,570. The next step is to round it to the nearest hundred dollars as the exercise requires. Since the tens digit is 7 (which is greater than 5), we increase the hundreds digit by 1, resulting in $28,600.
This step ensures our answer conforms to the desired specification of an easier to read number.