Problem 95 - 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 the United States, we spend an average of \(\$ 4000\) per person each year on health care- the highest in the world. What do we spend each year on health care nationwide? Express the answer in scientific notation.

Step-by-Step Solution

Answer

The total amount spent on healthcare in the U.S per year is approximately \(1.12 \times 10^{12}\) dollars.

Step 1: Identify the Population Figure

The population of the U.S. in 2000 is given as \(2.8 \times 10^{8}\).

Step 2: Identify the Amount Spent per Person on Health Care

The average amount spent per person on health care is \(\$ 4000\) per year or approximately \(4 \times 10^3\) in scientific notation.

Step 3: Multiply to Find the Total Amount

To find the total amount spent on health care in the U.S per year, multiply the population of the U.S by the average amount spent per person. The multiplication of these two quantities gives: \(2.8 \times 10^{8} \times 4 \times 10^{3} = 11.2 \times 10^{11}\). However, in correct scientific notation format, this equals \(1.12 \times 10^{12}\).

Key Concepts

Population Figure

The population figure refers to the estimated count of people living in a specific area at a given time. In the context of our exercise, the U.S. population in the year 2000 was estimated to be \(2.8 \times 10^{8}\). Understanding population figures is important for analyzing various national metrics, such as health care spending, education, and infrastructure needs.

Population figures are often given in scientific notation when dealing with very large numbers, like millions or billions of people. This makes the numbers easier to read and work with in calculations. For instance, \(2.8 \times 10^{8}\) means 280 million people, which gives a concise way to express large demographic data.

To interpret or use population figures correctly, consider:

• The unit - in this case, people.
• The time - this figure is from the year 2000.
• The context - it's related to calculating total health care spending.

Using scientific notation helps in simplifying otherwise cumbersome calculations that can arise in real-world scenarios.

Health Care Spending

Health care spending refers to the total expenses associated with services and products for maintaining or improving health. In our example, each person in the U.S. in 2000 spent, on average, \(\$4000\) annually on health care. This equates to \(4 \times 10^3\) when expressed in scientific notation.

Health care spending can be broken down into categories such as hospital care, physician services, and medication. Understanding these expenses on both an individual and national level is crucial for budget planning and policy making.

When evaluating health care spending in scientific notation, keep in mind:

• Conversion of monetary figures into scientific notation ensures precision in calculations.
• Comparing health care spending across different years or countries often involves large numbers, which are easier to manipulate in scientific notation form.
• Examining trends can highlight increases or decreases in spending relevant to economic, population, or health care advancements.

Scientific notation thus plays a crucial role in handling and understanding vast monetary amounts effectively.

Multiplication in Scientific Notation

Multiplying numbers presented in scientific notation involves a straightforward method that simplifies calculations, especially with large numbers. For this exercise, we needed to find the total annual health care expenditure in the U.S. by multiplying the population figure with the individual spending.

The initial multiplication is \(2.8 \times 10^{8}\) (the population) and \(4 \times 10^{3}\) (the spending per person). To multiply, follow these steps:

• Multiply the base numbers: \(2.8 \times 4 = 11.2\).
• Add the exponents for \(10\): \(8 + 3 = 11\).
• Combine to form \(11.2 \times 10^{11}\).

However, in scientific notation, the result should have one digit to the left of the decimal point. Therefore, adjust by moving the decimal one place to the left to get \(1.12\), and increase the exponent by one: \(11 + 1 = 12\), giving \(1.12 \times 10^{12}\).

Multiplication in scientific notation:

• Simplifies processing significantly large numbers.
• Reduces errors in extensive computations.
• Streamlines comparisons and understanding of national or global figures.

This makes it an essential tool in scientific and financial assessments.