Please help with a simple problem that entails vast and short periods of time, using equivalents and proportions.

[Retired Earthling]

I'm somewhat embarrassed I'm stuck on this. Plus I only have a basic calculator. I want to get a feel for the age and vastness of the universe, comparing the length of a human life (about 80 years) to different factors. For example, the age of the earth (4.5B years) beginning of hominoids (3M years) first Homo Sapiens (300K years). A spark is described as being .001 seconds. So for example if 80 years was the equivalent of a spark, what would be the equivalent number of years? How many sparks, or fractions of sparks, would be equivalent to each factor? I would appreciate too an explanation of methodology. Hopefully my explanation is clear enough. Thank you very much you mathematical wizards!

 

[topsquark]

I'm somewhat embarrassed I'm stuck on this. Plus I only have a basic calculator. I want to get a feel for the age and vastness of the universe, comparing the length of a human life (about 80 years) to different factors. For example, the age of the earth (4.5B years) beginning of hominoids (3M years) first Homo Sapiens (300K years). A spark is described as being .001 seconds. So for example if 80 years was the equivalent of a spark, what would be the equivalent number of years? How many sparks, or fractions of sparks, would be equivalent to each factor? I would appreciate too an explanation of methodology. Hopefully my explanation is clear enough. Thank you very much you mathematical wizards!

It looks like you are seeking a unit conversion.
[imath]\dfrac{ 4.5 \times 10^9 ~\text{yrs} }{ 1 } \cdot \dfrac{ 1 ~ \text{human lifetime} }{ 80 ~ \text{yrs} } = \dfrac{ 4.5 \times 10^9 \cdot \cancel{ \text{yrs} } \cdot \text{human lifetime} }{ 80 ~ \cancel{ \text{yrs} } } = \dfrac{ 4.5 \times 10^9 ~ \text{human lifetime} }{80} = 56250000 ~ \text{human lifetime}[/imath]

-Dan

 

[BigBeachBanana]

It looks like you are seeking a unit conversion.
[imath]\dfrac{ 4.5 \times 10^9 ~\text{yrs} }{ 1 } \cdot \dfrac{ 1 ~ \text{human lifetime} }{ 80 ~ \text{yrs} } = \dfrac{ 4.5 \times 10^9 \cdot \cancel{ \text{yrs} } \cdot \text{human lifetime} }{ 80 ~ \cancel{ \text{yrs} } } = \dfrac{ 4.5 \times 10^9 ~ \text{human lifetime} }{80} = 56250000 ~ \text{human lifetime}[/imath]

-Dan

Unit cancellation was how I passed physics and chemistry in school.

 

[topsquark]

Unit cancellation was how I passed physics and chemistry in school.

It's how I was taught to do stoichiometry in Chemistry. I liked the setup so much that's how I do all my unit conversions now.

-Dan

 

[Otis]

AKA: Dimensional Analysis. Very useful, visually as well as computationally.

?
[imath]\;[/imath]

 

[Otis]

A spark is … .001 seconds … if 80 years [were] the equivalent of a spark what would be the equivalent number of years?

Hi Retired Earthling. The phrases highlighted in red above have torqued my mind. It seems like you're talking about defining a new category of seconds and years because you've basically asked how many years is equivalent to 80 years.

The units 'year' and 'second' are human inventions. If we desire to invent a different kind of second or year, then I think we ought to use different names (eg: cosmic year versus human year).

There are 1,000 human sparks in a human second. Let's assume that one cosmic second contains 1,000 cosmic sparks, also.

If 1,000 cosmic sparks are each 80 human years long, then a cosmic second would be 80,000 human years long.

As there are 31,557,600 human seconds in a human year (accounting for leap years), a proportionate cosmic year would be 2,524,608,000,000 human years long.

Yet, that still doesn't seem (to me) what you've got in mind because you also asked:

How many sparks, or fractions of sparks, would be equivalent to each factor?

If you're simply changing the definition of 'spark' (as a thought experiment), then we don't need cosmic units, and topsquark's numerical result is the same as the number of (newly-defined) sparks in 4.5 billion years.

\(\displaystyle \dfrac{1\text{ spark}}{80\cancel{\text{ year}}} \cdot \dfrac{4500000000\cancel{\text{ year}}}{1} = 56250000\text{ spark}\)

I'm somewhat embarrassed

Me, too.


[imath]\;[/imath]

 

[Retired Earthling]

Hi Retired Earthling. The phrases highlighted in red above have torqued my mind. It seems like you're talking about defining a new category of seconds and years because you've basically asked how many years is equivalent to 80 years.

The units 'year' and 'second' are human inventions. If we desire to invent a different kind of second or year, then I think we ought to use different names (eg: cosmic year versus human year).

There are 1,000 human sparks in a human second. Let's assume that one cosmic second contains 1,000 cosmic sparks, also.

If 1,000 cosmic sparks are each 80 human years long, then a cosmic second would be 80,000 human years long.

As there are 31,557,600 human seconds in a human year (accounting for leap years), a proportionate cosmic year would be 2,524,608,000,000 human years long.

Yet, that still doesn't seem (to me) what you've got in mind because you also asked:


If you're simply changing the definition of 'spark' (as a thought experiment), then we don't need cosmic units, and topsquark's numerical result is the same as the number of (newly-defined) sparks in 4.5 billion years.

\(\displaystyle \dfrac{1\text{ spark}}{80\cancel{\text{ year}}} \cdot \dfrac{4500000000\cancel{\text{ year}}}{1} = 56250000\text{ spark}\)


Me, too.


[imath]\;[/imath]

Otis, You've torqued MY mind. Maybe I'm not able to articulate what I'm asking you to help me solve. Maybe the best word is equivalent. And ratio. For example, after I wrote this question I horsed around with it some more, and what I came up with is the following approximation: In attempting to illustrate the relationship between how long humans have walked the earth (300K years) and the age of the earth (4.5B years), (a very small fraction indeed) would yield an equivalent ratio between the length of a spark (.001 seconds) to a timeframe of 3.2M years. This might be totally off base, but could you, Dan, or any other you wizards come up with the correct answer? Or another problem would be the ratio of the length of a human life (80 years) and the aforesaid 330K of human existence? (one spark over 8 hours which I just made up). Thank you for any help!

 

[Deleted member 4993]

Otis, You've torqued MY mind. Maybe I'm not able to articulate what I'm asking you to help me solve. Maybe the best word is equivalent. And ratio. For example, after I wrote this question I horsed around with it some more, and what I came up with is the following approximation: In attempting to illustrate the relationship between how long humans have walked the earth (300K years) and the age of the earth (4.5B years), (a very small fraction indeed) would yield an equivalent ratio between the length of a spark (.001 seconds) to a timeframe of 3.2M years. This might be totally off base, but could you, Dan, or any other you wizards come up with the correct answer? Or another problem would be the ratio of the length of a human life (80 years) and the aforesaid 330K of human existence? (one spark over 8 hours which I just made up). Thank you for any help!

how long humans have walked the earth (300K years) and the age of the earth (4.5B years) = 300*10^3 / (4.5 * 10^9) (years/years) = ??

Actually, if you are really an old-timer - you will not need a calculator................