Problem 84 - Beginning and Intermediate Algebra

Question

One parsec is about 3.26 light-years and 1 light-year is about \(5.88 \times 10^{12}\) miles. Find the number of miles in 1 parsec.

Step-by-Step Solution

Answer

1 parsec is approximately equal to \(19.18 \times 10^{12}\) miles.

Step 1: Identify the Conversion Factors

We are given two conversion factors: 1 parsec is about 3.26 light-years, and 1 light-year is about \(5.88 \times 10^{12}\) miles. Step 2: Convert Parsecs to Light-Years

Step 2: Convert Parsecs to Light-Years

We need to find the number of miles in 1 parsec. To accomplish this, we will first convert parsecs to light-years. Since 1 parsec is equal to 3.26 light-years, we can simply multiply 1 parsec by this conversion factor: \[1 \,\text{parsec} \times 3.26 \,\text{light-years/parsec} = 3.26 \,\text{light-years}\] Step 3: Convert Light-Years to Miles

Step 3: Convert Light-Years to Miles

Now that we have converted parsecs to light-years, we will convert light-years to miles. We are given that 1 light-year is equal to \(5.88 \times 10^{12}\) miles. Therefore, multiply the number of light-years in 1 parsec (3.26 light-years) by the number of miles in 1 light-year (\(5.88 \times 10^{12}\) miles): \[3.26 \,\text{light-years} \times (5.88 \times 10^{12}) \,\text{miles/light-year}\] Step 4: Calculate the Number of Miles in One Parsec

Step 4: Calculate the Number of Miles in One Parsec

Perform the multiplication in step 3, and we find the number of miles in 1 parsec: \[3.26 \,\text{light-years} \times (5.88 \times 10^{12}) \,\text{miles/light-year} = (3.26 \times 5.88) \times 10^{12} \,\text{miles}\] \[(3.26 \times 5.88) \times 10^{12} \,\text{miles} \approx 19.18 \times 10^{12} \,\text{miles}\] So, 1 parsec is approximately equal to \(19.18 \times 10^{12}\) miles.

Key Concepts

Understanding Conversion Factors

Conversion factors are useful tools that make it easy to switch units from one measurement system to another. By understanding these factors, you can perform conversions smoothly to solve problems. In this exercise, two conversion factors are introduced:

• 1 parsec equals 3.26 light-years
• 1 light-year equals \(5.88 \times 10^{12}\) miles

To find out how many miles are in 1 parsec, you need to utilize both of these conversion factors. This stepwise conversion process helps simplify the problem.

Grasping Light-Years

A light-year is the distance that light travels in one year. Light moves fast, covering enormous distances quickly. This makes the light-year a handy unit for measuring distances in space. It's important to recall that light is the fastest thing in the universe, traveling at roughly 186,282 miles per second.In terms of the numbers, since 1 light-year equals \(5.88 \times 10^{12}\) miles, light travels an extraordinary distance in just a single year. Using this massive number as a conversion factor allows us to link astronomical units, like parsecs, to more measurable units, such as miles.

Interpreting Miles

Miles are a familiar unit of distance for many people, especially in countries like the United States. This unit brings a vast concept, such as the distance of a light-year, into a more earthly context. By converting astronomical distances into miles, we make these vast stretches of space more relatable and comprehendible. This conversion helps to bridge our earthly perspective with the immense scales encountered in space exploration. When we convert light-years or parsecs into miles, we connect something distant and abstract to a concrete and practical framework.

Approaching Algebraic Calculations

Algebraic calculations allow us to solve for unknowns by manipulating known values and equations. In this conversion problem, algebraic multiplication plays a key role in linking parsecs to miles.The process involves a series of steps:

• First, multiply the number of parsecs by the light-years per parsec (3.26 light-years).
• Then, multiply the resulting light-years by miles per light-year \((5.88 \times 10^{12})\).
• Perform these calculations: \(3.26 \times 5.88 \rightarrow 19.18\).

Finally, multiplying \(19.18\) by \(10^{12}\) results in approximately \(19.18 \times 10^{12}\) miles. This systematic approach underscores the power of simplification in solving complex astronomical problems.