Problem 106 - Elementary Algebra

Question

\(D=\left(\frac{6 \text { light-years }}{1}\right)\left(\frac{9.47 \times 10^{12} \mathrm{~km}}{1 \text { light-year }}\right)\left(\frac{1 \mathrm{mi}}{1.61 \mathrm{~km}}\right)\)

Step-by-Step Solution

Answer

3.53 x 10^{13} miles

Step 1: Identify the units and constants

Identify the units given in the problem. Here, the units are light-years, kilometers, and miles. Also identify their conversion factors: 1 light-year = 9.47 × 10^{12} km1 km = 1 / 1.61 miles

Step 2: Set up the dimensional analysis

Write the expression provided, ensuring that units cancel out appropriately:to calculate a product of fractions,include all the givenconversion factorsin the formula for an accurate computation of the total distance

Step 3: Convert light-years to kilometers

Multiply the given light-years by the conversion factor to change light-years to kilometers:distance in km = 6 light-years × 9.47 × 10^{12} km/light-year

Step 4: Convert kilometers to miles

Convert the result from kilometers to miles using the conversion factor 1 km = 1 / 1.61 miles:distance in miles = (6 × 9.47 × 10^{12} km) × (1 mile / 1.61 km)

Step 5: Perform calculations

Multiply the constants and perform the arithmetic operations step-by-step:6 × 9.47 × 10^{12} = 56.82 × 10^{12}dividing by 1.61 = (56.82 × 10^{12}) / 1.61 = 3.53 x 10^{13} miles

Key Concepts

Unit Conversion

Unit conversion is the process of changing values from one unit of measure to another. This is crucial in science and everyday calculations.
For example, converting distances, weights, and volumes can help us understand different measurements under one common system. To convert units accurately:

• Identify the original unit and the unit you want to convert to.
• Use the appropriate conversion factor (a ratio that expresses how many of one unit are equal to another).
• Multiply the quantity you're converting by the conversion factor.

In our exercise, we are converting light-years to kilometers and then kilometers to miles using specific conversion factors.

Light-Year to Kilometer Conversion

A light-year is the distance that light travels in one year. For scientific purposes, this unit helps measure vast interstellar distances. 1 light-year is equal to 9.47 × 10^{12} kilometers.
In our exercise:

• The given distance is 6 light-years.
• To convert to kilometers, use the factor: 9.47 × 10^{12} km/light-year.
• So, 6 light-years = 6 × 9.47 × 10^{12} = 56.82 × 10^{12} km.

This step simplifies vast distances into more comprehendible figures.

Kilometer to Mile Conversion

Kilometers and miles are commonly used to measure distances. The conversion factor between these units is:
1 kilometer = 1 / 1.61 miles. Here’s how to convert kilometers into miles:

• Find the number of kilometers you want to convert.
• Use the conversion factor: multiply the kilometers by 1 mile and then divide by 1.61.

For our exercise:

• Take the result from earlier: 56.82 × 10^{12} km.
• Convert to miles: 56.82 × 10^{12} km × (1 mile / 1.61 km) = 3.53 × 10^{13} miles.

This allows us to understand our distances in miles.

Arithmetic Operations

Performing arithmetic operations ensures accurate results in conversions. Understanding basic operations like multiplication and division is essential. Let’s break down the steps:

• First, identify and multiply all constants.
• Then, convert units by dividing or multiplying by the appropriate conversion factors.

For this problem:

• Convert 6 light-years to kilometers: 6 × 9.47 × 10^{12} = 56.82 × 10^{12} km.
• Then convert kilometers to miles by dividing: 56.82 × 10^{12} / 1.61 = 3.53 × 10^{13} miles.

Accurate arithmetic operations are key to achieving precise conversions.