Problem 58 - Contemporary Precalculus

Question

Express the given number in normal decimal notation. One light-year is the distance light travels in a 365 -day year. The speed of light is about 186,282.4 miles per second. (a) How long is 1 light-year (in miles)? Express your answer in scientific notation. (b) Light from the North Star takes 680 years to reach the earth. How many miles is the North Star from the earth?

Step-by-Step Solution

Answer

Question: Calculate the distance of 1 light-year in miles and express it in scientific notation. Then, calculate the distance of the North Star from the earth in miles and express it in scientific notation. Answer: The distance of 1 light-year is approximately 5.88 × 10^12 miles, and the distance of the North Star from the earth is approximately 4.00 × 10^15 miles.

Step 1: Calculate the distance of 1 light-year in miles

To find the distance of 1 light-year in miles, first, we need to convert a year to seconds. There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365 days in a year. So, we will multiply all these values to get the total number of seconds per year: seconds in a year = 60 × 60 × 24 × 365 Now, to find the distance of 1 light-year, multiply the speed of light by the number of seconds in a year: distance = speed × time distance = 186,282.4 miles/second × 60 × 60 × 24 × 365

Step 2: Express the distance of 1 light-year in scientific notation

Now, we need to express the result from Step 1 in scientific notation. The general form for scientific notation is a × 10^b, where 1 ≤ a < 10 and b is an integer. Calculate the distance and then convert the result to scientific notation: distance ≈ 5,878,625,373,184 miles Therefore, the distance of 1 light-year in scientific notation is approximately: 5.88 × 10^12 miles.

Step 3: Calculate the distance of the North Star from the earth in miles

We know that 1 light-year is approximately 5.88 × 10^12 miles. The light from the North Star takes 680 years to reach the earth. So, to find the distance of the North Star from the earth, we need to multiply the distance of 1 light-year by 680 years: distance = (1 light-year distance) × (time in light-years) distance = (5.88 × 10^12 miles) × 680

Step 4: Express the distance of the North Star from the earth in scientific notation

Now, we need to express the result from Step 3 in scientific notation. Calculate the distance and then convert the result to scientific notation: distance ≈ 3,997,925,253,728,000 miles Therefore, the distance of the North Star from the earth in scientific notation is approximately: 4.00 × 10^15 miles.

Key Concepts

Light-Year Calculations

A light-year is a way to measure distance, specifically how far light can travel in one year. Since the speed of light is constant, we can calculate how far it goes in a certain period. To find how long one light-year is in miles, we first need to determine how many seconds are in a year. This entails multiplying the number of seconds in a minute, minutes in an hour, hours in a day, and days in a year:
- 60 seconds/minute - 60 minutes/hour - 24 hours/day - 365 days/year

When you multiply these numbers, you get the total seconds per year. With this number, you multiply it by the speed of light (186,282.4 miles per second) to find the distance light travels in one year. That gives you the distance of one light-year. Expressing this vast number in scientific notation makes it more manageable to work with, especially in astronomical calculations.

Distance of Stars

The distance of stars from Earth is often measured in light-years because these distances are extremely large, and using miles or kilometers would result in cumbersome numbers. For example, the North Star is approximately 680 light-years away from Earth. To find the distance in miles, we multiply the number of light-years by the distance of one light-year (calculated as 5.88 × 10^12 miles).

This multiplication gives us a large number that can be expressed in scientific notation as 4.00 × 10^15 miles. Using light-years makes conversations about astronomical distances both easier and more relatable when describing how far away various celestial objects are.

Speed of Light

The speed of light is a constant and crucial figure in physics, especially in astronomy. It measures approximately 186,282.4 miles per second. This rapid speed helps dictate how we understand distance and time in space. Because light travels so fast, we use it as a measure of distance known as a light-year, which represents how far light travels in one year.

Understanding the speed of light allows for sophisticated calculations in the study of the universe and helps astronomers determine distances between celestial bodies efficiently. For instance, when we say an astronomical object is "X light-years" away, it means light from that object has taken X years to reach us. This provides a concrete way to think about vast distances in space.

Unit Conversion

Unit conversion is essential in scientific calculations, especially those involving astronomical distances. This process allows us to convert units from one form to another, such as from seconds to years, or from miles to kilometers. In the context of light-year calculations, we convert time units (seconds, minutes, hours, days, and years) to find out how many seconds are in a year.

Similarly, expressing large numbers in scientific notation allows for easier calculations and comparisons. For instance, instead of using trillions of miles, we use the scientific notation form (like 5.88 × 10^12 miles for one light-year) to keep numbers easier to read and manipulate. This method helps scientists and students alike handle astronomical data efficiently.