Problem 83 - Beginning and Intermediate Algebra

Question

The nearest star, Alpha Centauri C, is about 4.22 light-years from Earth. One light-year is the distance that light travels in 1 year and is about \(5.88 \times 10^{12} \mathrm{mi}\) How many miles is it from Earth to Alpha Centauri C?

Step-by-Step Solution

Answer

The distance from Earth to Alpha Centauri C is approximately \(2.48 \times 10^{13}\) miles.

Step 1: Identify the given information

We know the following information: - Distance between Earth and Alpha Centauri C (in light-years): 4.22 light-years - One light-year: \(5.88 \times 10^{12}\) miles

Step 2: Set up the distance conversion equation

The distance (in miles) can be expressed as follows: Distance (in miles) = Distance (in light-years) × miles per light-year

Step 3: Plug in the given information and compute

Using the given information, we can plug the values into the equation: Distance (in miles) = 4.22 light-years × \(5.88 \times 10^{12}\) miles per light-year

Step 4: Calculate the distance in miles

Multiply the distance (in light-years) by the number of miles per light-year: Distance (in miles) = \(4.22 \times 5.88 \times 10^{12}\) miles

Step 5: Simplify the expression

Calculate the expression to find the distance (in miles): Distance (in miles) ≈ \(2.48 \times 10^{13}\) miles Thus, the distance from Earth to Alpha Centauri C is approximately \(2.48 \times 10^{13}\) miles.

Key Concepts

Light-Year Calculations

When we talk about astronomical distances, we often use light-years. A light-year measures how far light travels in one year. Since light moves extremely fast, about 186,282 miles per second, it ends up traveling an incredibly large distance over the span of a year. To put it simply, one light-year equals the distance light covers in 365 days of continuous, uninterrupted travel.
For practical purposes in astronomy, this distance is approximately 5.88 trillion miles. This number, however large it might seem, provides a manageable way to express and work with the vast distances between stars.
So, if a star is 4.22 light-years away from Earth, it means that it would take light 4.22 years to get here from there. This makes it easier for astronomers to handle and understand the immense distances in the universe.

Multiplication of Scientific Notation

To manage extremely large or small numbers more easily, we use scientific notation. In scientific notation, numbers are expressed as a product of a number between 1 and 10 and a power of 10. For instance, the number 5.88 trillion can be conveniently written as \(5.88 \times 10^{12}\).
Multiplying numbers in scientific notation allows us to handle astronomically large numbers without the fuss. Here's how it works:

• First, multiply the decimal parts, which are numbers between 1 and 10. For our problem, it's 4.22 and 5.88.
• Second, multiply the powers of ten by adding their exponents together.

Let's apply this to the exercise: we multiply 4.22 and 5.88 to get a product of 24.8136.
Then, since we are multiplying by \(10^{12}\), we simply tack on this power of ten to result in \(24.8136 \times 10^{12}\).
However, scientific notation standards often require expressing the result as \(2.48136 \times 10^{13}\) by adjusting the decimal place. This slight adjustment makes it easier to standardize our answer.

Unit Conversion in Astronomy

Astronomy often deals with massive and mind-boggling distances. Thus, converting units such as light-years to miles is essential for contextual understanding. Unit conversion in astronomy involves taking a known distance in one unit and expressing it in another unit. This ensures that calculations are practical and understandable.
For our problem, converting light-years to miles requires us to multiply the number of light-years by the equivalent distance in miles per light-year. In this case, we use:

• 4.22 light-years
• \(5.88 \times 10^{12}\) miles per light-year

By performing the multiplication, we convert the distance into a more relatable unit, which results in approximately \(2.48 \times 10^{13}\) miles.
This practice allows astronomers and students alike to discuss the vastness of space in terms we can connect with more directly, offering clearer and more intuitive understanding of distances that are otherwise abstract.