Miles In Light-Year

[harpazo]

Section R.2
Algebra Essentials
Michael Sullivan
Textbook: College Algebra Edition 9

One light-year is defined by astronomers to be the distance that a beam of light will travel in 1 year or 365 days. If the speed of light is 186,000 miles/second, how many miles are in a light-year? Express your answer in scientific notation.

Solution:

I know from reading the question that light travels at 186,000 miles for each second.
There are 3600 seconds in one hour. I then proceed to multiply 186,000 by 3600 seconds to get 669,600,000 miles in one hour.

One day has 24 hours. So,
669,600,000 x 24 hours is
16,070,400,000 miles in one day.
So, for 365 days in one year, I multiply 16,070,400,000 x 365 days to get 5,865,696,000,000 miles in a year.

In scientific notation, the answer is expressed as follows:

5.86 x 10^12 miles per year.

Is this right? Did I mess up somewhere along the way? If there is an easier set up or equation, please share that with me here.

 

[Otis]

… If there is an easier set up …

Only you can decide what's easier for you, but here's a different set-up.

\(\displaystyle \frac{186000 \text{ mile}}{\cancel{\text{ second}}} \times \frac{60×60×24×365 \cancel{\text{ second}}}{\text{year}}\)



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[Dr.Peterson]

One day has 24 hours. So,
669,600,000 x 24 hours is
16,070,400,000 miles in one day.
So, for 365 days in one year, I multiply 16,070,400,000 x 365 days to get 5,865,696,000,000 miles in a year.

In scientific notation, the answer is expressed as follows:

5.86 x 10^12 miles per year.

Check your rounding.

 

[harpazo]

Only you can decide what's easier for you, but here's a different set-up.

\(\displaystyle \frac{186000 \text{ mile}}{\cancel{\text{ second}}} \times \frac{60×60×24×365 \cancel{\text{ second}}}{\text{year}}\)



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Unit conversions?

 

[harpazo]

Check your rounding.

5.87 x 10^12 miles per year?

 

[Dr.Peterson]

5.87 x 10^12 miles per year?

Yes, if they asked for three significant digits, or you recognize that that is appropriate. (The book may or may not be emphasizing that.)

It's worth pointing out that a year is 365 days to three significant digits, but not exactly that, due to leap years.

 

[harpazo]

Yes, if they asked for three significant digits, or you recognize that that is appropriate. (The book may or may not be emphasizing that.)

It's worth pointing out that a year is 365 days to three significant digits, but not exactly that, due to leap years.

Does the answer change by much for a leap year?

 

[Dr.Peterson]

The issue is not that it changes in a leap year (a unit of length can't depend on what year it is!), but that the "year" used in the definition of a light year is not the length of this year or that year, but of a "standard year" in a particular sense, which is 365.25 days, not 365. See Wikipedia:

As defined by the IAU, the light-year is the product of the Julian year (365.25 days as opposed to the 365.2425-day Gregorian year) and the speed of light (299792458 m/s [= 186282.397 miles per second]).​

Using these more exact numbers in your calculation, we get 5,878,625,371,567, which rounds to 5.88*10^12 miles.

Of course, all this is irrelevant to your problem, for which you have to use the numbers you were given. It's just a point of interest.

 

[harpazo]

The issue is not that it changes in a leap year (a unit of length can't depend on what year it is!), but that the "year" used in the definition of a light year is not the length of this year or that year, but of a "standard year" in a particular sense, which is 365.25 days, not 365. See Wikipedia:

As defined by the IAU, the light-year is the product of the Julian year (365.25 days as opposed to the 365.2425-day Gregorian year) and the speed of light (299792458 m/s [= 186282.397 miles per second]).​

Using these more exact numbers in your calculation, we get 5,878,625,371,567, which rounds to 5.88*10^12 miles.

Of course, all this is irrelevant to your problem, for which you have to use the numbers you were given. It's just a point of interest.

Thank you for the information. Check out your PM. I responded to your answers.

 

[Otis]

Unit conversions?

Yes. Post #2 shows one way to convert from the unit miles-per-second to the unit miles-per-year.

We call the ratio on the right the 'conversion factor', and its numerical value is 1 (as Jomo pointed out) because the numerator and denominator both represent the same thing. That is, 60×60×24×365 seconds is the same amount of time as 1 year.

Note 1: In post #2, I could have typed those denominators as "1 second" and "1 year", but that's not necessary. When we don't see a numerical value in front of a unit, then it's understood to be 1.

Note 2: Instead of combining individual conversion factors into a single factor, I could have written out multiple conversion factors (seconds/hour, hours/day, days/year), similar to but not exactly the same as what Dr. Peterson showed you before. The way I did it in post #2 is a shortcut.

PS: If you're interested in yet another approach, you can review how you handled the exercise last May.

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